Frobenius manifolds and moduli
spaces for singularities
CLAUS HERTLING
Cambridge
UNIVERSITY PRESS

CAMBRIDGE TRACTS IN MATHEMATICS
General Editors
B. BOLLOBAS, W. FULTON, А. КАТОК, F. KIRWAN, P. SARNAK
151 Frobenius manifolds and
moduli spaces for singularities

Contents
Preface . page viii
Part 1. Multiplication on the tangent bundle
1 Introduction to part 1 3
1.1 First examples 4
1.2 Fast track through the results 5
2 Definition and first properties of F-manifolds 9
2.1 Finite-dimensional algebras 9
2.2 Vector bundles with multiplication 11
2.3 Definition of F-manifolds 14
2.4 Decomposition of F-manifolds and examples 16
2.5 F-manifolds and potentiality 19
3 Massive F-manifolds and Lagrange maps 23
3.1 Lagrange property of massive F-manifolds 23
3.2 Existence of Euler fields 26
3.3 Lyashko-Looijenga maps and graphs of Lagrange maps 29
3.4 Miniversal Lagrange maps and F-manifolds 32
3.5 Lyashko-Looijenga map of an F-manifold 35
4 Discriminants and modality of F-manifolds 40
4.1 Discriminant of an F-manifold 40
4.2 2-dimensional F-manifolds 44
4.3 Logarithmic vector fields 47
ix 4.4 Isomorphisms and modality of germs of F-manifolds 52
\\ 4.5 Analytic spectrum embedded differently 56

vi Contents
5 Singularities and Coxeter groups 61
5.1 Hypersurface singularities 61
5.2 Boundary singularities 69
5.3 Coxeter groups and F-manifolds 75
5.4 Coxeter groups and Frobenius manifolds 82
5.5 3-dimensional and other F-manifolds 87
Part 2. Frobenius manifolds, GauB-Manin connections, and
moduli spaces for hypersurface singularities
6 Introduction to part 2 99
6.1 Construction of Frobenius manifolds for singularities 100
6.2 Moduli spaces and other applications 104
7 Connections over the punctured plane 109
7.1 Flat vector bundles on the punctured plane 109
7.2 Lattices 113
7.3 Saturated lattices 116
7.4 Riemann-Hilbert-Birkhoff problem 120
7.5 Spectral numbers globally 128
8 Meromorphic connections 131
8.1 Logarithmic vector fields and differential forms 131
8.2 Logarithmic pole along a smooth divisor 134
8.3 Logarithmic pole along any divisor 139
8.4 Remarks on regular singular connections 143
9 Frobenius manifolds and second structure connections 145
9.1 Definition of Frobenius manifolds 145
9.2 Second structure connections 148
9.3 First structure connections 154
9.4 From the structure connections to metric and multiplication 157
9.5 Massive Frobenius manifolds 160
10 GauB-Manin connections for hypersurface singularities 165
10.1 Semiuniversal unfoldings and F-manifolds 165
10.2 Cohomology bundle 167
10.3 GauB-Manin connection 170
10.4 Higher residue pairings 179
10.5 Polarized mixed Hodge structures and opposite filtrations 183
10.6 Brieskorn lattice 188
Contents
11 Frobenius manifolds for hypersurface singularities
11.1 Construction of Frobenius manifolds
11.2 Deformed flat coordinates
11.3 Remarks on mirror symmetry
11.4 Remarks on oscillating integrals
12 д-constant stratum
12.1 Canonical complex structure
12.2 Period map and infinitesimal Torelli
13 Moduli spaces for singularities
13.1 Compatibilities
13.2 Symmetries of singularities
13.3 Global moduli spaces for singularities
14 Variance of the spectral numbers
14.1 Socle field
14.2 G-function of a massive Frobenius manifold
14.3 Variance of the spectrum
Bibliography
Index
vn
195
195
205
211
212
218
218
224
230
230
235
240
248
248
251
256
260
269

Preface
Frobenius manifolds are complex manifolds with a rich structure on the holo-
morphic tangent bundle, a multiplication and a metric which harmonize in the
most natural way. They were defined by Dubrovin in 1991, motivated by the
work of Witten, Dijkgraaf, E. Verlinde, and H. Verlinde on topological field
theory. Originally coming from physics, Frobenius manifolds now turn up in
very different areas of mathematics, giving unexpected relations between them,
in quantum cohomology, singularity theory, integrable systems, symplectic ge-
geometry, and others. The isomorphy of certain Frobenius manifolds in quantum
cohomology and in singularity theory is one version of mirror symmetry.
This book is devoted to the relations between Frobenius manifolds and sin-
singularity theory. It consists of two parts.
In part 1 F-manifolds are studied, manifolds with a multiplication on the
tangent bundle with a natural integrability condition. They were introduced in
[HM][Man2, I§5]. Frobenius manifolds are F-manifolds. Studying F-manifolds,
one is led directly to discriminants, a classical subject of singularity theory, and
to Lagrange maps and their singularities. Our development of the general struc-
structure of F-manifolds is at the same time an introduction to discriminants and
Lagrange maps. As an application, we use some work of Givental to prove a
conjecture of Dubrovin about Frobenius manifolds and Coxeter groups.
In part 2 we take up the construction of Frobenius manifolds in singularity
theory. Already in 1983 K. Saito and M. S aito had found that the base space of a
semiuniversal unfolding of an isolated hypersurface singularity can be equipped
with the structure of a Frobenius manifold. Their construction involves the
GauB-Manin connection, polarized mixed Hodge structures, K. Saito's higher
residue pairings, and his primitive forms. It was hardly accessible for nonspe-
cialists. We give a more elementary detailed account of the construction, explain
all ingredients, and develop or cite all necessary results.
via
Preface
IX
We give a number of applications. The deepest one is the construction of
global moduli spaces for isolated hypersurface singularities.
The construction of K. Saito and M. Saito is related to a recent construction
of Frobenius manifolds via oscillating integrals by Sabbah and Barannikov. We
comment upon that.
Background and other books. The reader should know the basic concepts
of complex analytic geometry, including coherent sheaves and flatness (cf. for
example [Fi]). All notions from symplectic geometry which are used can be
found in [AGV1, chapter 18]. An excellent basic reference on flat connections
and vector bundles (and much more) is the forthcoming book [Sab4]. It also
gives a viewpoint on Frobenius manifolds which complements ours. Two fun-
fundamental books on Frobenius manifolds are [Du3] and [Man2]. Our treatment
of singularities and their GauB-Manin connection is essentially self-contained
and gives precisely what is needed, but it is quite compact. Some books which
expound several aspects in much more detail are [AGVl][AGV2][Ku][Lo2].
Acknowledgements. This book grew out of my habilitation. I would like to
thank many people. E. Brieskorn was my teacher in singularity theory and
defined in 1970 the wonderful object Щ', which is now called the Brieskorn
lattice. Yu. Manin introduced me to Frobenius manifolds. The common paper
[HM] was the starting point for part 1. His papers and those of B. Dubrovin
and C. Sabbah, and discussions with them were very fruitful. G.-M. Greuel and
G. Pfister sharpened my view of moduli problems. M. Schulze and M. Rosellen
made useful comments.
Of course, this book builds on the work of many people in singularity the-
theory; Arnold, Givental, Looijenga, Malgrange, K. Saito, M. Saito, Scherk, O.P.
Shcherbak, Slodowy, Steenbrink, Teissier, Varchenko, Wall, and many others.
A good part of the book was written during a stay at the mathematics depart-
department of the University Paul Sabatier in Toulouse. I thank the department and
especially J.-F. Mattel for their hospitality.
Bonn, July 2001
Claus Hertling

Parti
Multiplication on the tangent bundle

Chapter 1
Introduction to part 1
An F-manifold is a complex manifold M such that each holomorphic tangent
space TtM,t e M, is a commutative and associative algebra with unit element,
and the multiplication varies in a specific way with the point t e M. More
precisely, it is a triple (M, о, ё) where о is an O^-bilinear commutative and
associative multiplication on the holomorphic tangent sheaf Тм, е is a global
unit field, and the multiplication satisfies the integrability condition
LieXoy(o) = X о Liey(o) + Y о Liex(o) A.1)
for any two local vector fields X and Y in Тм. This notion was first defined
in [HM][Man2, I§5], motivated by Frobenius manifolds. Frobenius manifolds
are F-manifolds.
Part 1 of this book is devoted to the local structure of F-manifolds. It turns
out to be closely related to singularity theory and symplectic geometry. Discri-
Discriminants and Lagrange maps play a key role.
In the short section 1.1 of this introduction the reader can experience some
of the geometry of F-manifolds. We sketch a construction of 2-dimensional
F-manifolds which shows how F-manifolds turn up 'in nature' and how they
are related to discriminants. In section 1.2 we offer a fast track through the main
notions and results of chapters 2 to 5.
In chapters 2 to 4 the general structure of F-manifolds is developed. In
chapter 5 the most important classes of F-manifolds are discussed.
In chapter 2 F-manifolds are defined and some basic properties are estab-
established. One property shows that F-manifolds decompose locally in a nice way.
Another one describes the relation to connections, metrics, and the potentiality
condition of Frobenius manifolds.
In chapter 3 the relation to symplectic geometry and especially to Lagrange
maps is discussed. This allows use to be made of Givental's paper [Gi2] on
singular Lagrange varieties and their Lagrange maps.

4 Introduction to part I
Chapter 4 presents several notions and results, which are mostly motivated
by corresponding notions and results in singularity theory. Most important are
the discriminants and their geometry.
In chapter 5 F-manifolds from hypersurface singularities, boundary singu-
singularities, and Coxeter groups are discussed. In the case of Coxeter groups we
extend some results of Givental [Gi2] and use them to prove a conjecture of
Dubrovin about their Frobenius manifolds.
The reader should have the following background. There should be fa-
familiarity with the basic concepts of complex analytic geometry, including
coherent sheaves and flatness. One reference is [Fi]. There should also be
awareness of those notions from symplectic geometry which are treated in
[AGV1, chapter 18] (canonical 1-form on the cotangent bundle, Lagrange
fibration, Lagrange map, generating function). We recommend this reference. In
chapter 5 some acquaintance with singularity theory makes the reading easier,
but it is not necessary. Good references are [AGV1] and [Lo2].
1.1 First examples
To give the reader an idea of what F-manifolds look like and how they arise
naturally, a construction of 2-dimensional F-manifolds is sketched. A systematic
treatment is given in sections 4.1 and 4.2.
Let W be a finite Coxeter group of type him), m > 2, acting on R2 and (by
C-linear extension) on C2. Then the ring С [jci,jc2]1v С C[x\,X2]oiW -invariant
polynomials is C[jc] ,x{\w = C[fi, ti\ with 2 homogeneous generators t\ and
t2 of degrees m and 2. Therefore the quotient space C2/ W =: M is isomorphic
to C2 as an affine algebraic variety, and the vector field e := ^- is unique up to
multiplication by a constant. The image in M of the union of the complexified
reflection hyperplanes is the discriminant V. We choose t\ and t-i such that it is
given as V = {t e M \ t\ - ^ff = 0}.
For a point t e M with Гг ф 0, the pair (e, V) gives rise to a multiplication
on T,M in the following way, which is illustrated in figure 1.1.
The e-orbit through the point t intersects the discriminant at 2 points. We
shift the tangent hyperplanes of V at these points with the flow of e to TtM.
We find that they are transverse to one another and to e. Therefore there are 2
unique vectors e\ and e% in T,M which are tangent to these lines and satisfy
е = е\+в2- We define a multiplication on T, M by e,- о ej := 5y e,-. It is obviously
commutative and associative, and e is the unit vector.
If we write this multiplication in terms of the coordinate fields e :— ?-
and ^-, after some calculation we find 4г о 4г = *о~2 ¦ irr> and e is the
Ot2 0/2 O?2 ^ 3/i '
1.2 Fast track through the results
Figure 1.1
unit field. Therefore the multiplication extends holomorphically to the whole
tangent bundle TM. One can show that it satisfies A.1). The orbit space M is
an F-manifold.
This construction of an F-manifold from a discriminant V and a transver-
transversal vector field e extends to higher dimensions (Corollary 4.6) and yields
F-manifolds in many other cases, for example for all finite Coxeter groups
(section 5.3).
1.2 Fast track through the results
The most notable (germs of) F-manifolds with many typical and some special
properties are the base spaces of semiuniversal unfoldings of isolated hyper-
hypersurface singularities and of boundary singularities (sections 5.1 and 5.2). Here
the tangent space at each parameter is canonically isomorphic to the sum of the
Jacobi algebras of the singularities above this parameter. Many of the general
results on F-manifolds have been known in another guise in the hypersurface
singularity case and all should be compared with it.
One reason why the integrability condition A.1) is natural is the following:
Let (M, p) be the germ of an F-manifold (M, o, e). The algebra TPM decom-
decomposes uniquely into a sum of (irreducible) local algebras which annihilate one
another (Lemma 2.1). Now the integrability condition A.1) ensures that this
infinitesimal decomposition extends to a unique decomposition of the germ
(M, p) into a product of germs of F-manifolds (Theorem 2.11).
If the multiplication at TpM is semisimple, that is, if TpM decomposes into
1 -dimensional algebras, then this provides canonical coordinates и \,..., un on
(M, p) with g|: о g2_ = 5^. _i-. in fact, at points with semisimple multiplication
the integrability condition A.1) is equivalent to the existence of such canonical
coordinates. In the hypersurface case, the decomposition of the germ (M, p)

6 Introduction to part 1
for some parameter p is the unique decomposition into a product of base spaces
of semiuniversal unfoldings of the singularities above p.
Another reason why A.1) is natural is its relation to the potentiality of
Frobenius manifolds. There exist F-manifolds such that not all tangent spaces
are Frobenius algebras. They cannot be Frobenius manifolds. But if all tangent
spaces are Frobenius algebras then the integrability condition A.1) is related
to a version of potentiality which requires a metric on M that is multiplication
invariant, but not necessarily flat. See section 2.5 for details.
The most important geometric object which is attributed to an n-dimensional
manifold M with multiplication о on the tangent sheaf Тм and unit field e
(with or without A.1)) is the analytic spectrum L := Specan(T^) с T*M
(see section 2.2). The projection ж : L -> M is flat and finite of degree n.
The fibre n~\p) С L above p e M consists of the set Homaig(TpM, C) of
algebra homomorphisms from Tp M to C; they correspond 1 -1 to the irreducible
subalgebras of (TpM, o) (see Lemma 2.1). The multiplication on TM can be
recovered from L, because the map
a : T
M
X н> a(X)\L
A.2)
is an isomorphism of CV-algebras; here X is any lift of X to T*M and a is the
canonical 1-form on T*M. The values of the function a(X) on л~1(р) are the
eigenvalues of Xo : TPM -» TpM.
The analytic spectrum Lisa reduced variety if and only if the multiplication
is generically semisimple. Then the manifold with multiplication (M, o, e) is
called massive. Now, a third reason why the integrability condition A.1) is
natural is this: Suppose that (M, o, e) is a manifold with generically semisimple
multiplication. Then L с T*M is a Lagrange variety if and only if (M, o, <?) is
a massive F-manifold (Theorem 3.2).
The main body of part 1 is devoted to the study of germs of massive
F-manifolds at points where the multiplication is not semisimple.
We will make use of the theory of singular Lagrange varieties and their
Lagrange maps, which has been worked out by Givental in [Gi2]. In fact, the
notion of an irreducible germ (with respect to the above decomposition) of
a massive F-manifold is equivalent to Givental's notion of a miniversal germ
of a flat Lagrange map (Theorem 3.16). Via this equivalence Givental's paper
contains many results on massive F-manifolds and will be extremely useful.
Locally the canonical 1-form a on T*M can be integrated on the ana-
analytic spectrum L of a massive F-manifold (M, o, e) to a generating function
F : L -? С which is continuous on L and holomorphic on Lreg. It depends on
a property of L, which is weaker than normality or maximality of the complex
structure of L, whether F is holomorphic on L (see section 3.2).
1.2 Fast track through the results 7
If F is holomorphic on L then it corresponds via A.2) to an Euler field
E — a~\F) of weight 1, that is, a vector field on M with Lie?(o) = о
(Theorem 3.3).
In any case, a generating function F : L -> С gives rise to a Lyashko-
Looijenga map Л : M -» C" (see sections 3.3 and 3.5) and a discriminant
V=tt(F-](O))cM.
If F is holomorphic and an Euler field E = a~](F) exists then the discrimi-
discriminant V is the hypersurface of points where the multiplication with E is not in-
vertible.ThenitisafreedivisorwithlogarithmicfieldsDer/M(log'P) = EoTM
(Theorem 4.9). This generalizes a result of K. Saito for the hypersurface case.
From the unit field e and a discriminant V С M one can reconstruct every-
everything. One can read off the multiplication on TM in a very nice elementary
way (Corollary 4.6 and section 1.1): The e-orbit of a generic point p e M
intersects V at n points. One shifts the n tangent hyperplanes with the flow of
e to TpM. Then there exist unique vectors e\(p),..., en(p) e TPM such that
?"=i ei(P) = e(p) and YH=\ c • е<(^) = tpm and such Ла1 the subspaces
?\ ,. С • e,{p), j = 1,..., n, are the shifted hyperplanes. The multiplication
on TPM is given by <?,(/?) о ej(p) = 3,7e,-(p).
In the case of hypersurface singularities and boundary singularities, the clas-
classical discriminant in the base space of a semiuniversal unfolding is such a
discriminant. The critical set С in the total space of the unfolding is canonically
isomorphic to the analytic spectrum L; this isomorphism identifies the map a
in A.2) with a Kodaira-Spencer map ac : TM -> (пс)*Ос and a generating
function F : L —> С with the restriction of the unfolding function to the critical
set C. This Kodaira-Spencer map ac is the source of the multiplication on TM
in the hypersurface singularity case. The multiplication on TM had first been
defined in this way by K. Saito.
Critical set and analytic spectrum are smooth in the hypersurface singular-
singularity case. By the work of Arnold and Hormander on Lagrange maps and sin-
singularities an excellent correspondence holds (Theorem 5.6): each irreducible
germ of a massive F-manifold with smooth analytic spectrum comes from an
isolated hypersurface singularity, and this singularity is unique up to stable right
equivalence.
By the work of Nguyen huu Due and Nguyen tien Dai the same corre-
correspondence holds for boundary singularities and irreducible germs of massive
F-manifolds whose analytic spectrum is isomorphic to (C1, 0) x ({(*, у) б
C2 | xy = 0}, 0) with ordered components (Theorem 5.14).
The complex orbit space M := C/ W = C" of a finite irreducible Coxeter
group W carries an (up to some rescaling) canonical structure of a massive
F-manifold: A generating system P\,...,Pn of W-invariant homogeneous

8
Introduction to part 1
polynomials induces coordinates (| („onM. Precisely one polynomial,
e.g. Pi, has highest degree. The field щ is up to a scalar independent of any
choices. This field e := ^ as the unit field and the classical discriminant
V С M, the image of the reflection hyperplanes, determine in the elementary
way described above the structure of a massive F-manifold. This follows from
[Du2][Du3, Lecture 4] as well as from [Gi2, Theorem 14].
Dubrovin established the structure of a Frobenius manifold on the complex
orbit space M = C/ W, with this multiplication, with K. Saito's flat metric on
M, and with a canonical Euler field with positive weights (see Theorem 5.23). At
the same place he conjectured that these Frobenius manifolds and their products
are (up to some well-understood rescalings) the only massive Frobenius man-
manifolds with an Euler field with positive weights. We will prove this conjecture
(Theorem 5.25).
Crucial for the proof is Givental's result [Gi2, Theorem 14]. It characterizes
the germs (M, 0) of F-manifolds of irreducible Coxeter groups by geometric
properties (see Theorem 5.21). We obtain from it the following intermediate
result (Theorem 5.20): An irreducible germ (M, p) of a simple F-manifold such
that Tp M is a Frobenius algebra is isomorphic to the germ at 0 of the F-manifold
of an irreducible Coxeter group.
A massive F-manifold (M, o, e) is called simple if the germs (M, p), p € M,
of F-manifolds are contained in finitely many isomorphism classes. Theorem
5.20 complements in a nice way the relation of irreducible Coxeter groups
to the simple hypersurface singularities An, Dn, En and the simple boundary
singularities Bn,Cn,F4.
In dimensions 1 and 2, up to isomorphism all the irreducible germs of massive
F-manifolds come from the irreducible Coxeter groups Ai and I2(m) (m > 3)
with /2C) = A2, /2D) = B2, /2E) =: Я2, /2F) = G2. But already in dime-
dimension 3 the classification is vast (see section 5.5).
Chapter 2
Definition and first properties of F-manifolds
An F-manifold is a manifold with a multiplication on the tangent bundle which
satisfies a certain integrability condition. It is defined in section 2.3. Sections
2.4 and 2.5 give two reasons why this is a good notion. In section 2.4 it is
shown that germs of F-manifolds decompose in a nice way. In section 2.5 the
relation to connections and metrics is discussed. It turns out that the integrability
condition is part of the potentiality condition for Frobenius manifolds. Therefore
Frobenius manifolds are F-manifolds.
Section 2.1 is a self-contained elementary account of the structure of finite
dimensional algebras in general (e.g. the tangent spaces of an F-manifold)
and Frobenius algebras in particular. Section 2.2 discusses vector bundles with
multiplication. There the caustic and the analytic spectrum are defined, two
notions which are important for F-manifolds.
2.1 Finite-dimensional algebras
In this section (Q, o, e) is a C-algebra of finite dimension (> 1) with commuta-
commutative and associative multiplication and with unit e. The next lemma gives precise
information on the decomposition of Q into irreducible algebras. The statements
are well known and elementary. They can be deduced directly in the given order
or from more general results in commutative algebra (Q is an Artin algebra).
Algebra homomorphisms are always supposed to map the unit to the unit.
Lemma 2.1 Let (Q, o, e) be as above. As the endomorphisms xo : Q -» Q,
x € Q, commute, there is a unique simultaneous decomposition Q = ф^=1 Qk
into generalized eigenspaces Qk (with dimc Qk > U Define ek e Qk by e =
El=i ek- Then
(i) One has Qj о Qk = Ofor j ф к; also ek ф 0 and ej о ек = 8]кек; the
element ek is the unit of the algebra Qk = ek о Q.

10
Definition and first properties of F-manifolds
(ii) The function Xk : Q -» С which associates to x € Q the eigenvalue of xo
on Qk is an algebra homomorphism; Xj ф Xkforj ф к.
(iii) The algebra (Qk, o, ek) is an irreducible and a local algebra with maximal
ideal mk = Qk П ker(A.*).
(iv) The subsets ker(Xt) = m^ ф 0^ Qj,k = l,..,l, are the maximal ideals
of the algebra Q; the complement Q — \Jk ker(A.*) is the group ofinvertible
elements of Q.
(v) Theset{Xu...,Xi}^ Homc_a^(<2, C).
(vi) The localization <2кег(щ is isomorphic to Qk.
We call this decomposition the eigenspace decomposition of (Q,o,e).
The set L := {X\,..., А./} С Q* carries a natural complex structure Ol such
that OL{L) = Q and 0/.,^ = ?>*. More details on this will be given in
section 2.2.
The algebra (or its multiplication) is called semisimple if Q decomposes into
1-dimensional subspaces, Q = 0^ Qk = ®f™te С • ek.
An irreducible algebra Q = С • e ф m with maximal ideal m is a Gorenstein
ring if the socle AnnG(m) has dimension 1.
An algebra Q = 0^=1 Qk is a Frobenius algebra if each irreducible sub-
algebra is a Gorenstein ring (cf. for example [Kun]).
The next (also well known) lemma gives equivalent conditions and additional
information. Note that this classical definition of a Frobenius algebra is slightly
weaker than Dubrovin's: he calls an algebra (Q, o, e) together with a. fixed
bilinear form g as in Lemma 2.2 (a) (iii) a Frobenius algebra.
Lemma 2.2 (a) The following conditions are equivalent.
(i) The algebra (Q, o, e) is a Frobenius algebra,
(ii) As a Q-module Hom(<2, C) = Q.
(iii) There exists a bilinear form g : Q x Q -> С which is symmetric, non-
degenerate and multiplication invariant, i.e. g(a о b, c) = g(a, b о с).
(b) Let Q = 0j=| Qk be a Frobenius algebra and Qk = С • ek ф тк. The
generators of Hom(?>, C) as a Q-module are the linear forms f:Q-*-C with
/(AnnGt (щО) = С for all k.
One obtains a 1-1 correspondence between these linear forms and the bi-
bilinear forms g as in (a) (iii) by putting g(x, y) := f(x о у).
Proof: (a) Any of the conditions (i), (ii), (iii) in (a) is satisfied for Q if and
only if it is satisfied for each irreducible subalgebra Qk. One checks this with
Qj ° Qk = 0 for j' ф к. So we may suppose that Q is irreducible.
2.2 Vector bundles with multiplication
11
(i) <;—>¦ (ii) A linear form /eHomB,C) generates HomB,C) as a
g-module if and only if the linear form (x i->- f(y о х)) is nontrivial for
any у € Q - {0}, that is, if and only if f(y о Q) - С for any у e Q - {0}.
The socle AnnQ(m) is the set of the common eigenvectors of all endomor-
phismsxo : Q -> Q, x € Q.Ifdim Ann2(m) > 2 then for any linear form / an
element у 6 (ker / П Ann2(m)) - {0} satisfies у о Q — С • у and f(y о Q) = 0;
so / does not generate Hom(<2, C). If dim Ann2(m) = 1 then it is contained
in any nontrivial ideal, because any such ideal contains a common eigenvector
of all endomorphisms. The set у о Q for у € Q - {0} is an ideal. So, then a
linear form / with /(Апп2(т)) = С generates HomB, C) as a Q-module.
(i) =$¦ (iii) Choose any linear form / with /(AnnG(m)) = С and define g
by g(x, y) ."= f(x о у). It remains to show that g is nondegenerate. But for any
x € Q - {0} there exists а у € Q with С • x о у = AnnG(m), because AnnG(m)
is contained in the ideal xo Q.
(iii) =»(i) The equalities g(m, Ann2(m)) = g(e,moAnn2(m)) = g(e, 0) = 0
imply dim Anne(m) = 1.
(b) Starting with a bilinear form g, the corresponding linear form / is given
by f(x) = g(x, e). The rest is clear from the preceding discussion. Q
The semisimple algebra Q = (Bf™® С • e^is a Frobenius algebra.
A classical result is that the complete intersections Сс™,о/(/ь • • •. fm) are
Gorenstein. But there are other Gorenstein algebras, e.g. C{x, y, z}/(x2, y2,xz,
yz, xy — z2) is Gorenstein, but not a complete intersection.
Finally, in the next section vector bundles with multiplication will be consid-
considered. Condition (ii) of Lemma 2.2 (a) shows that there the points whose fibres
are Frobenius algebras form an open set in the base.
2.2 Vector bundles with multiplication
Now we consider a holomorphic vector bundle Q —> M on a complex manifold
M with multiplication on the fibres: The sheaf Q of holomorphic sections of the
bundle Q -> M is equipped with an См-bilinear commutative and associative
multiplication о and with a global unit section e.
The set (JpeM HomC-aig(Q(p), C) of algebra homomorphisms from the sin-
single fibres Q(p) to С (which map the unit to 1 6 C) is a subset of the dual bundle
Q* and has a natural complex structure. It is the analytic spectrum Specan(Q).
We sketch the definition ([Hou, ch. 3], also [Fi, 1.14]):
The См-sheaf SymOM Q can be identified with the Cw-sheaf of holomorphic
functions on Q* which are polynomial in the fibres. The canonical CM-algebra
homomorphism SymOM Q -*¦ Q which maps the multiplication in SymOM Q

12
Definition and first properties of F-manifolds
to the multiplication о in Q is surjective. The kernel generates an ideal I
in OQ..
One can describe the ideal locally explicitly: suppose U С М is open and
&\,..., &„ e Q([/)is abase of sections of the restriction Q | v -> U with 5, = e
and S; о Sj = ?t af,<5b denote by y\,..., yn the fibrewise linear functions on
Q*\U which are induced by <5i,..., <5n; then the ideal I is generated in Q*\U by
-1 and
B.1)
The support of Oq.IT with the restriction of OQ./2 as structure sheaf is
Specan(Q) С Q*. We denote the natural projections by jzq. :Q*-+M and
7Г : Specan(Q) -*¦ M. A part of the following theorem is already clear from
the discussion. A complete proof and thorough discussion can be found in
[Hou, ch. 3].
Theorem 2.3 The set \JpeM Homc_afe (?)(», C) is the support of the analytic
spectrum Specan(Q) =: L. The composition of maps
a : Q *-+ {tzq.)*Oq. -> 7t»OL
B.2)
is an isomorphism of О M-algebras and of free О м -modules of rank n, here n
is the fibre dimension ofQ-*M. The projection n : L -*¦ M is finite and flat
of degree n.
Consider a point p e M with eigenspace decomposition Q(p)= ®l=i Qk(p)
and L П 7r~' (p) = {Х\,..., ХцР)}. The restriction of the isomorphism
Up)
to the fibre over p yields isomorphisms
Qk(p) = OL
С
B.3)
B.4)
Corollary 2.4 In a sufficiently small neighbourhood U of a point p e M, the
eigenspace decomposition Q(p)= ф'^, Qk(p) of the fibre Q(p) extends
uniquely to a decomposition of the bundle Q\y —> U into multiplication in-
invariant holomorphic subbundles.
Proof of Corollary 2.4: The OMp-free submodules OLikk in the decomposition
in B.3) of (я*Оь)р are obviously multiplication invariant. Via the isomorphism
a one obtains locally a decomposition of the sheaf Q of sections of Q -*¦ M
into multiplication invariant free CV-submodules. D
2.2 Vector bundles with multiplication
13
Of course, the induced decomposition of Q (q) for a point q in the neighbour-
neighbourhood of p may be coarser than the eigenspace decomposition of Q(q).
The base is naturally stratified with respect to the numbers and dimensions of
the components of the eigenspace decompositions of the fibres of Q ->¦ M. To
make this precise we introduce a partial ordering >- on the set V of partitions of n:
!««
p = (Pi,..., Pm) | Pi e N, Pi > pi+i, J^Pi =
for P, у е V define
3or:{l,...,/(y)}-Ml,...,/(/*)} s.t Pj = J^ y,.
The partition P(p) of a fibre Q(p) is the partition of n = dim Q(p) by the
dimensions of the subspaces of the eigenspace decomposition.
Proposition 2.5 Fix a partition P e V. The subset {p e M | P{p) >- P\ is
empty or an analytic subset of M.
Proof. The partition P{f) of a polynomial of degree n is the partition of n by
the multiplicity of the zeros of /.
Fact: The space {a e C" | P(zn + ?"=i щг"~') >¦ P) is an algebraic subvariety
of С with normalization isomorphic to C'w.
For the proof one only has to regard the finite map C->-C\ иь>
((-1)'<Т|(и))|=1,..,п (сГ|(") is the i-th symmetric polynomial).
A section X &Q(U),V CM open, induces via the coefficients of the charac-
characteristic polynomial pch,xo of multiplication by X a holomorphic map U -*¦ C.
Hence the set {q e U | P(pch,Xo) > P\ is analytic. The intersection of such
analytic sets for a basis of sections in U is {q e [/ | P(q) >- P). ?
Suppose that M is connected. Then there is a unique partition Po such that
(p 6 M | P(p) = j8o) is open in M. The complement К := [p 6 M \ P{p) ф
Po\ will be called the caustic in M; this name is motivated by the Lagrange
maps (sections 3.1, 3.3, 3.4) and the hypersurface singularities (section 5.1).
The multiplication is generically semisimple if and only if/Jo = (l,...,l).
Proposition 2.6 The caustic K, is a hypersurface or empty.
Proof. Locally in M — K, there is a holomorphically varying decomposition
Q(p) = ф^=1 Qk(p) with partial unit fields e\,..., e/.
Suppose dim(X], p) < dim M -2 for some point p e K.. Then in a neighbour-
neighbourhood U the complement U — K. is simply connected. There is no monodromy

14
Definition and first properties of F-manifolds
for the locally defined vector fields e\,..., et in U - K.. They extend to vector
fields in U. For p e U — K. the map e^o : Q(p)-+ Q{p) is the projec-
projection to Qk(p)- Because of e\ + ¦ ¦ ¦ + <?/ = e these projections extend to all
of U and yield a decomposition of Q(p) as above for all p € U. Hence
/С П U = 0. ?
2.3 Definition of F-manifolds
An F-manifold is a manifold Л/ with a multiplication on the tangent bundle
TM which harmonizes with the Lie bracket in the most natural way. A neat
formulation of this property requires the Lie derivative of tensors.
Remark 2.7 Here the sheaf of (k, /)-tensors (k, I e No) on a manifold M is
the sheaf of C>A/-module homomorphisms HomoM@*=1 TM, ®|=1 Тм). А
@, /)-tensor T : OM -*¦ ®|=, TM can be identified with 7"A). Vector fields are
@, l)-tensors, a (commutative) multiplication on TM is a (symmetric) B,1)-
tensor.
The Lie derivative Liex with respect to a vector field X is a derivation on the
sheaf of (k, Z)-tensors. It is Liex(/) = X(f) for functions /, Uex(Y) = [X, Y]
for vector fields Y, Uex(Yi ® ... ® Я) = ?,- F] <g> ..[X, F;].. <g> Y, for @, /)-
tensors, and (Liex7")(F) = Liex(T(Y))- T(Liex(Y))for(k, /)-tensors T. One
can always write it explicitly with Lie brackets. Because of the Jacobi identity
the Lie derivative satisfies Lie^y] == [Liex, Liey].
Definition 2.8 (a) An F-manifold is a triple (M, o, e) where M is a com-
complex connected manifold of dimension > 1, о is a commutative and associative
Ом-bilinear multiplication Тм х Тм -*¦ Тм, е is a global unit field, and the
multiplication satisfies for any two local vector fields X, Y
Liew(o) = X о Liey(o) + Y о Liex(o).
B.5)
(b) Let (M, o, e) be an F-manifold. An Euler field E of weight d € С is a
global vector field E which satisfies
Lie?(o) = d ¦ o. B.6)
(If no weight is mentioned, an Euler field will usually mean an Euler field of
weight 1.)
Remarks 2.9 (i) We do not require that the algebras (TpM,o,e(p)) are
Frobenius algebras (cf. section 2.1). Nevertheless, this is a distinguished class.
Frobenius manifolds are F-manifolds [HM][Manl, I§5]. This will be discussed
in section 2.5.
2.3 Definition of F-manifolds
15
(ii) Definition 2.8 differs slightly from the definition of F-manifolds in [HM]
by the addition of a global unit field e. This unit field is important, for example,
for the definition of SpecanG^/). Also, the Euler fields were called weak Euler
fields in [HM] in order to separate them from the Euler fields with stronger
properties of Frobenius manifolds. This is not necessary here.
(iii) Formula B.5) is equivalent to
[X о Y, Z о W] - [X о Y, Z] о W - [X о Y, W] о Z
- Y о [X, Z о W] + Y о [X, Z] о W + Y о [X, W] о Z = 0 B.7)
for any four (local) vector fields X, Y, Z, W. Formula B.6) is equivalent to
[E, X о Y] - [E, X] о Y - X о [E, Y] - d ¦ X о Y = 0 B.8)
for any two (local) vector fields X, Y. The left hand side of B.8) is OM-
polylinear with respect to X and Y, because Liee(o) is a B, l)-tensor. The
left hand side of B.7) is Ом-polylinear with respect to X, Y, Z, W. Hence it
defines a D, l)-tensor. In order to check B.5) and B.6) for a manifold with
multiplication, it suffices to check B.7) and B.8) for a basis of vector fields.
(iv) The unit field e in an F-manifold (M, o, e) plays a distinguished role. It
is automatically nowhere vanishing. It is an Euler field of weight 0,
Liee(o) = 0-o,
B.9)
because of B.5) for X = F = e. So, the multiplication of the F-manifold is
constant along the unit field.
(v) An Euler field E of weight d ф 0 is not constant along the unit field. But
one has for any d eC
[e,E] = d-e, B.10)
because of B.8) for X = F = e. More generally, in [HM][Manl, I§5] the
identity
[E°n, Eom] = d(m - n) ¦
B.11)
is proved. Section 3.1 will show how intrinsic the notion of an Euler field is for
an F-manifold.
(vi) The sheaf of Euler fields of an F-manifold (M, o, e) is a Lie subalgebra
of TM. If E{ and E2 are Euler fields of weight d\ and d2, then с ¦ E\ (c € C)
is an Euler field of weight с • d\, E\ + Ег is an Euler field of weight d\ + d2,
and [Ei, E2] is an Euler field of weight 0. The last holds because of Lie[?,?2] =
[Lie?l, Lie?J (cf. Remark 2.7 and [HM][Manl, I§5]).

16
Definition and first properties of F-manifolds
(vii) The caustic К of an F-manifold is the subvariety of points p e M such
that the eigenspace decomposition of TpM has fewer components than for
generic points (cf. section 2.2). The caustic is invariant with respect to e because
of B.9).
2.4 Decomposition of F-manifolds and examples
Proposition 2.10 The product oftwo F-manifolds (M\, ou e\)and(M2, o2, e2)
is an F-manifold (M, o, e) = (M\ x M2, o\ ф о2, e\ + e2).
If E\ and Ei are Eider fields on M\ and M2 of the same weight d then the
sum E\ + E2 (of the lifts to M) is an Eider field of weight d on M.
Proof. The tangent sheaf decomposes,
Та = OM ¦ pr^TMl © OM ¦ pr^TMl.
Any vector fields Xit У, e prJ~lTMl, i = 1, 2, satisfy
X, о Y, e prr'TMi, [X,, Y,] e Pr7lTMl,
XioY2 = 0, [XuY2] = 0.
This together with B.7) for vector fields in Тщ and for vector fields in Тмг gives
B.7) for vector fields in prj"" lTMl U pr^T^. Because of the O^-polylinearity
then B.7) holds for all vector fields. For the same reasons, E\ + E2 satisfies
B.8). D
Theorem 2.11 Let (M, p) be the germ in p e M of an F-manifold (M, o, e).
Then the eigenspace decomposition TpM — ®'k=i(TpM)k of the algebra
Tp M extends to a unique decomposition
1
(M, p) = Y\(Mk, p)
k=\
ofthegerm(M, p) into a product of germs of F-manifolds. These germs (Mk, p)
are irreducible germs of F-manifolds, as the algebras TpMk = (TpM)k are
already irreducible.
An Euler field E on (M, p) decomposes into a sum of Eulerfields of the same
weights on the germs (Mk, p) of F-manifolds.
Proof. By Corollary 2.4 the eigenspace decomposition of TpM extends in some
neighbourhood of p to a decomposition of the tangent bundle into a sum of
2.4 Decomposition of F-manifolds and examples
17
multiplication invariant subbundles. First we have to show that these subbundles
and any sum of them are integrable.
Accordingly, let TMtP = ®k=i(TMtP)k be the decomposition of TM<p into
multiplication invariant free 0MiP-submodules, and e = J^k ek with ek е
(TM,P)k- Then eko : TM<p ->¦ (TMiP)k is the projection; ey oek= 8jkek.
Claim:
(i) With respect to ek the multiplication is invariant, Lieet(o) = 0 • o;
(ii) the vectorfields ej and ek commute, [ej, ek] = 0;
(Hi) they leave the subsheaves invariant, [ej, (TMtP)k] С (ТМ:Р)к;
(iv) the subsheaves satisfy [(TMiP)j, (Тм,Р)к\ С (JmiP)j + (TM,p)k-
Proof of the claim: (i) The equality
Sjk ¦ Lieet(o) = Lieejoet(o) = e,- о Lie?l(o) + ek о Lieej(o).
implies for j ф к as well as for j = к that e,- о Liee4(o) = 0 ¦ o. Thus
(ii) The equality
0 = Lieej.(o)(eb ek) — [ej, ek о ek] — 2ek о [ej, ek~\
shows that [ejt ek] e (TMtP)k, so for j фкые have [e;-, e*] - 0, for j = к this
holds anyway.
(iii) Suppose X = ekoX e (TMiP)k; then
0 = Liee.(o)(eb X) = [ey, X]-eko [es, X].
(iv) Suppose X e (Тм>р);, У e (TMiP)k, кф1ф j; then e,- о X = 0 and
0 = Liee/oX(o)(e,, Y) = e,- о Liex(o)(e,-, Y)
= e,- о [X, et о У] - e,- о [X, e,-] о У - e,- о [X, У] о ef
= -е,- о [X, Y]. 0
Claim (iv) shows that for any к the subbundle with germs of sections
©;Y*C7m,p) j is integrable. According to the Frobenius theorem (cf. for example
[War, p. 41]) there is a (germ of a) submersion fk : (M, p) ->¦ (Cdim(r*M>', 0)
such that the fibres are the integral manifolds of this subbundle. Then ф^ fk :
(M, p) -*¦ (CdimM, 0) is an isomorphism.
The submanifolds (Mk, p) := (@;#t /;)~40), p) yield the decomposition
(M, p) = ni=i(^*- P) wi*

18
Definition and first properties of F-manifolds
Claim:
(v) If X, Y e prt-%4i/, then X о Y e pr^TMttP.
(vi) If E is an Euler field then eko E € prk~xTMkP.
Proof of the claim: (v) The product X о Y is contained in (9M>/) ¦ prk~
because this sheaf is multiplication invariant. Now X о Y e prk~lTMktP
if and only if [Z, X о Y] e {TM,p)j for any j and any Z e {TMtP)y, but
[Z, W]= Liez(X о Y) = UtejoZ{X о У)
= ej о Liez(X о Y) e (TM,p)j.
(vi) Analogously, for any Z € (TMiP)j
-[Z, ekoE]= LieetoE(ej о Z)
= Lieeto?(°)(e/, Z) + Lieeto?(e/) о Z + LieetO?-(Z) о е,-
E{ej) oZ + UeetoE(Z) о е,- е (TM
xTMk<p
is true
Claim (v) and (vi) show that the multiplication on (M, /?) and an Euler field
E come from multiplication and vector fields on the submanifolds (M^, p) via
the decomposition. These satisfy B.7) and B.8): this is just the restriction of
B.7) and B.8) to pr^TMktP.
?
Examples 2.12 (i) The manifold M = С with coordinate и and unit field e = ~-
аи
with multiplication e о e = e is an F-manifold. The field ? = u-e = M^-is
an Euler field of weight 1. The space of allEuler fields of weight d is d ¦ E + С ¦ e.
One has only to check B.7) and B.8) for X - Y = Z = W = e and compare
B.10).
Any 1-dimensional F-manifold is locally isomorphic to an open subset of
this F-manifold (С, о, e). It will be called A\.
(ii) From® and Proposition 2.10 one obtains the F-manifold A" — (С, о, e)
with coordinates u\,...,un, idempotent vector fields e-, = j^, semisimple
multiplication <?, о ej — 5,7e,-, unit field e = ?\ et and an Euler field E =
J2t и; • et of weight 1. Because of Theorem 2.11, the space of Euler fields of
weight d is d ¦ E + ?,. С • et.
Also because of Theorem 2.11, any F-manifold M with semisimple multi-
multiplication is locally isomorphic to an open subset of the F-manifold A". The
induced local coordinates u\,..., un on M are unique up to renumbering and
shift. They are called canonical coordinates, following Dubrovin. They are the
eigenvalues of a locally defined Euler field of weight 1.
2.5 F-manifolds and potentiality
19
(iii) Any Frobenius manifold is an F-manifold [HM][Manl, I§5], see
section 2.5.
(iv) Especially, the complex orbit space of a finite Coxeter group carries
the structure of a Frobenius manifold [Du2][Du3, Lecture 4]. The F-manifold
structure will be discussed in section 5.3, the Frobenius manifold structure
in section 5.4. Here we only give the multiplication and the Euler fields for
the 2-dimensional F-manifolds him), m > 2, with /2B) = A2, /2C) = A2,
/2D) = B2 = C2, /2E) =: Hi, /2F) = G2.
The manifold is M — C2 with coordinates t\, t2; we denote 5,- := jf. Unit
field e and multiplication о are given by e = Si and S2 о S2 = t2 ¦ S\. An
Euler iield E of weight 1 is E = tiSi + ^t2S2. The space of global Euler fields
of weight «5? is d ¦ E + С • e for m > 3. The caustic is 1С = {t e M \ t2 = 0} for
m > 3 and 1С = 0 for m = 2. The multiplication is semisimple outside of 1С;
the germ {M, t) is an irreducible germ of an F-manifold if and only if t € 1С.
One can check all of this by hand. We will come back to it in Theorem 4.7,
when more general results allow more insight.
(v) Another 2-dimensional F-manifold is C2 with coordinates tt, t2, unit field
e = 8] and multiplication о given by S2 о S2 = 0. Here all germs (M, t) are
irreducible and isomorphic. E\ := fiSi is an Euler field of weight 1. Contrary
to the above cases with generically semisimple multiplication, here the space
of Euler fields of weight 0 is infinite dimensional, by B.8):
{E | Lie?(o) = 0 • o}
= [E | [S\, E] =0, o2 о \b2, t\ = 0}
= {e,5, + s2(t2)S2 | e, б С, е2 б СЫС2), 5,(e2) = 0}. B.12)
(vi) The base space of a semiuniversal unfolding of an isolated hypersurface
singularity is (a germ of) an F-manifold. The multiplication was defined first by
K. Saito [SK6, A.5)] [SK9, A.3)]. A good part of the geometry of F-manifolds
that will be developed in the next sections is classical in the case of hypersurface
singularities, from different points of view. We will discuss this in section 5.1.
(vii) Also the base of a semiuniversal unfolding of a boundary singularity
is (a germ of) an F-manifold, compare section 5.2. There are certainly more
classes of semiuniversal unfoldings which carry the structure of F-manifolds.
2.5 F-manifolds and potentiality
The integrability condition B.5) for the multiplication in F-manifolds and
the potentiality condition in Frobenius manifolds are closely related. For
semisimple multiplication this has been known previously (with Theorem 3.2

20
Definition and first properties of F-manifolds
2.5 F-manifolds and potentiality
21
(i) ¦<=>¦ (ii) and e.g. [Hi, Theorem 3.1]). Here we give a general version,
requiring neither semisimple multiplication nor flatness of the metric.
Remarks 2.13 (a) In this section we need some basic notions from differential
geometry: connections, covariant derivative of vector fields, torsion freeness,
metric, Levi-Civita connection. For the real C°°-case one finds these in any text-
textbook on differential geometry. The translation to the complex and holomorphic
case here is straightforward.
(b) Let M be a manifold with a connection V. The covariant derivative
VXT of a (k, /)-tensor with respect to a vector field is defined exactly as the
Lie derivative Lie* Г in Remark 2.7, starting with the covariant derivatives of
vector fields. The operator Vx is a derivation on the sheaf of {k, /)-tensors just
as Lie*. But Vx is also OM-linear in X, opposite to Lie*. Therefore VT is a
(k + 1,0-tensor.
Theorem 2.14 Let (M, o, V) be a manifold M with a commutative and as-
associative multiplication о on TM and with a torsion free connection V. By
definition, V о (X, Y, Z) is symmetric in Y and Z.
If the C, \)-tensor Vo is symmetric in all three arguments, then the multipli-
multiplication satisfies for any local vector fields X and Y
LieXoy(o) = X о Liey(o) + Y о Liex(o).
B.13)
Proof. The term V о (X, Y, Z) = VX(F о Z) - VX(Y) oZ -Yo VX(Z) is
symmetric in Y and Z. The D, l)-tensor
(X, Y, Z, W) h> V о (X, Y о Z, W) + W о V о (X, Y, Z)
= VX(Y о Z о W) - VX(Y) oZoW
-Yo VX(Z) oW -YoZo V* (IV) B.14)
is symmetric in Y, Z, W. A simple calculation using the torsion freeness of V
shows
(LieXoK(o) - X о Liey(o) -Yo Liex(o))(Z, W)
= Vo(XoY,Z,W)-XoVo(Y,Z,W)-YoVo(X,Z,W)
-Vo(ZoW,X,Y) + ZoVo(W,X,Y)+WoVo(Z,X, Y).
B.15)
If V о is symmetric in all three arguments then
V о (X о Y, Z, W) + Z о V о (W, X, Y) + W о V о (Z, X, Y) B.16)
is symmetric in X, Y, Z, W because of the symmetry of the tensor in B.14).
Then the right hand side of B.15) vanishes. ?
Theorem 2.15 Let (M, o, e, g) be a manifold with a commutative and asso-
associative multiplication о on TM, a unit field e, and a metric g (a symmetric
nondegenerate bilinear form) on TM which is multiplication invariant, i.e. the
Q,0)-tensor A,
A(X, Y, Z) := g(X, Y о Z),
B.17)
is symmetric in all three arguments. V denotes the Levi-Civita connection of
the metric. The coidentity e is the I—form which is defined by e(X) = g(X, e).
The following conditions are equivalent:
(i) The manifold with multiplication and unit (M, o, e) is an F-manifold and
s is closed.
(ii) The D, 0)-tensor VA is symmetric in all four arguments.
(Hi) The C, \)-tensor Vo is symmetric in all three arguments.
Proof. The Levi-Civita connection satisfies Vg = 0. Therefore
VA(X, Y, Z, IV) = Xg(Y, Z о IV) - g(VxF, ZoW)
-g(Y,WoVxZ)-g(Y,ZoVxW)
= g(Y, VX(Z о IV) - W о VXZ - Z о VXW)
= g(Y, V о (X, Z, IV)). B.18)
The metric g is nondegenerate and VA(X, Y, Z, IV) is always symmetric in
Y, Z, W. Equation B.18) shows (ii) <=» (iii). Because of Vg = 0 and the
torsion freeness VXF — VyX = [X, Y], the 1-form s satisfies
de(X, Y) = X(e(Y)) - Y(s(X)) - e([X, Y])
= g(Y, Vxe) - g(X, VYe)
= -VA(X, Y, e, e) + VA(F, X, e, e).
B.19)
Hence (ii) =» de = 0; with Theorem 2.14 this gives (ii) =» (i). It remains to
show (i) => (ii).
The equations B.20) and B.21) follow from the definition of Vo and from
B.18),
V о (X, Y, e) = Y о V о (X, e, ё),
VA(X, U, Y, e) = VA(X, U о Y, e, e).
B.20)
B.21)

22 Definition and first properties of F-manifolds
One calculates with B.15) and B.18)
g(e, (Uexovio) -Xo Liey(o) - У о Lie* (o))(Z, W))
= VA(X о Y, e, Z, W) - VA(Y, X, Z, W) - VA(X, Y, Z, W)
- VA(Z о W, e, X, Y) + VA(W, Z, X, Y) + VA(Z, W, X, Y). B.22)
If (i) holds then B.19), B.21), and B.22) imply
VA(Y, X, Z, W) - VA(W, Z, X, Y)
= -VA{X, Y, Z, W) + VA(Z, W, X, Y). B.23)
The left hand side is symmetric in X and Z, the right hand side is skewsymmetric
in X and Z, so both sides vanish. VA is symmetric in all four arguments. D
Lemma 2.16 Let (M, g, V) be a manifold with metric g and Levi-Civita con-
connection V. Then a vector field Z is flat, i.e. VZ = 0, if and only if Liez(g) = 0
and the 1-form ez := g(Z,.) is closed.
Proof. The connection V is torsion free and satisfies Vg = 0. Therefore
(cf. B.19))
dez(X, Y) = g{Y, VXZ) - g(X, Vy Z),
, Y) = g(Y, VXZ) + g(X, VrZ).
B.24)
B.25)
?
Remarks 2.17 (a) Let (M, o, e, g) satisfy the hypotheses of Theorem 2.15 and
let g be flat. Then condition (ii) in Theorem 2.15 is equivalent to the existence
of a local potential Ф 6 OM,P (for any p e M) with (ATZ)O = A(X, У, Z)
for any flat local vector fields X, Y, Z.
(b) In view of this the conditions (ii) and (iii) in Theorem 2.15 are called
potentiality conditions.
(c) The manifold (M, о, е, Е, g) is a Frobenius manifold if it satisfies the
hypotheses and conditions in Theorem 2.15, if g is flat, if Liee(g) = 0 (respec-
(respectively e is flat), and if ? is an Euler field (of weight 1, with respect to M as
F-manifold), with UeE(g) = D ¦ g for some DeC (cf. Definition 9.1).
Chapter 3
Massive F-manifolds and Lagrange maps
In this section the relation between F-manifolds and symplectic geometry is
discussed. The most crucial fact is shown in section 3.1: the analytic spectrum
of a massive (i.e. with generically semisimple multiplication) F-manifold M is a
Lagrange variety L С T*M; and a Lagrange variety L С Т*М in the cotangent
bundle of a manifold M supplies the manifold M with the structure of an
F-manifold if and only if the map a : Тм —*¦ л* Oi from C.1) is an isomorphism.
The condition that this map a : TM -*¦ n*OL is an isomorphism is close to
Givental's notion of a miniversal Lagrange map [Gi2, ch. 13]. In section 3.4 the
correspondence between massive F-manifolds and Lagrange maps is rewritten
using this notion.
If E is an Euler field in a massive F-manifold M then the holomorphic func-
function F := a~'(?) : L —>• С satisfies dF\Lreg = a\Lreg (here a is the canonical
1-form on T*M). But as L may have singularities, the global existence of E
and of such a holomorphic function is not clear. This is discussed in section 3.2.
Much weaker than the existence of ? is the existence of a continuous function
F : L -> С which is holomorphic on Lreg withdF\Lreg = a\Lreg. This is called
a generating function for the massive F-manifold. It gives rise to the three notions
bifurcation diagram, Lyashko-Looijenga map, and discriminant. They turn out
to be holomorphic even if F is only continuous along Lsmg. This is discussed in
section 3.5 for F-manifolds and in section 3.3 more generally for Lagrange maps.
A good reference for the basic notions from symplectic geometry which
are used in this chapter (Lagrange variety, Lagrange fibration, Lagrange map,
generating function) is [AGV1, ch. 18].
3.1 Lagrange property of massive F-manifolds
Consider an n-dimensional manifold (M, o, e) with commutative and asso-
associative multiplication on the tangent bundle and with unit field e. Its analytic
23

24
Massive F-manifolds and Lagrange maps
spectrum L :— Specan(T^) is a subvariety of the cotangent bundle T*M. The
cotangent bundle carries a natural symplectic structure, given by the 2-form
da. Here a is the canonical 1-form, which is written as a = ?(. y,df, in local
coordinates t\ ,...,?„ for the base and dual coordinates y\,... ,yn for the fibres
Um -*• (Лт-MhUT'M, щ i-> yi).
The isomorphism a : TM -> я» OL from B.2) can be expressed with a by
C.1)
where X eTM and X is any lift of X to a neighbourhood of L in T*M.
The values of the function a(X) on л~1(р) are the eigenvalues of Xo on
T^Af; this follows from Theorem and Lemma 2.1.
Definition 3.1. A manifold (M, o, e) with commutative and associative mul-
multiplication on the tangent bundle and with unit field e is massive if the multipli-
multiplication is generically semisimple.
Then the set of points where the multiplication is not semisimple is empty
or a hypersurface, which is the caustic 1С (Proposition 2.6). In the rest of the
paper we will study the local structure of massive F-manifolds at points where
the multiplication is not semisimple.
It is known that the analytic spectrum of a massive Frobenius manifold is
Lagrange (compare [Au] and the references cited there). Theorem 3.2 to-
together with Theorem 2.15 make the relations between the different conditions
transparent.
Theorem 3.2 Let (M, o, e) be a massive n-dimensional manifold M.
The analytic spectrum L = SpecanG]y) с Т*М is an everywhere reduced
subvariety. The map л : L -*¦ M is finite and flat. It is a branched covering
of degree n, branched above the caustic 1С. The following conditions are equiv-
equivalent.
(i) The manifold (M, o, e) is a massive F-manifold;
(ii) At any semisimple point p e M — 1С, the idempotent vector fields e\,...,
en б Тм,р commute.
(Hi) The subvariety L С Т*М is a Lagrange variety, i.e. a\Lreg is closed.
Proof. The variety L— ж~1AС) is smooth, and л : L — л~1AС) -*¦ M — /Cisa
covering. The sheaf п„Оъ (= Тм) is a free 0M-module, so a Cohen-Macaulay
Ом-module and a Cohen-Macaulay ring. Then L is Cohen-Macaulay and
everywhere reduced, as it is reduced at generic points (cf. [Lo2, pp. 49-51] for
the notion Cohen-Macaulay and details of these arguments).
3.1 Lagrange property of massive F-manifolds
25
It remains to show the equivalences (i) <==>¦ (ii) ¦?=> (iii).
(i) => (ii) follows from Theorem 2.11 and has been discussed in Example
2.12 (ii).
(ii) =>• (i) is clear because B.8) holds everywhere if it holds at generic points
(in fact, one point suffices).
(ii) => (iii) We fix canonical coordinates м,- with ^ — e,- on (M, p) for a point
p 6 M — 1С and the dual coordinates x,- on the fibres of the cotangent bundle
G^ _>. {пТ'м)*От'м, ej h> Xj). Then locally above (M, p) the analytic spec-
spectrum L is in these coordinates
L = {(xj, uj) | Xi. -I \-xn = l,
П
= {J{(Xj,Uj)\xJ=8ij}.
C.2)
1=1
The 1-form a = J^Xjdu, is closed in L — л ]AС). This set is open and dense
in Lreg, hence L is a Lagrange variety.
(iii) => (ii) Above a small neighbourhood U of p e M - K, the analyticjpec-
trum consists of и smooth components L*, к = 1,... ,n, with ж : L^ -—>¦ U.
An idempotent vector field et can be lifted to vector fields e~i in neighbourhoods
Uk of Lk in T*M such that they are tangent to all Lk. The commutator [el, e'j]
is a lift of the commutator [et, ej] in these neighbourhoods Uk.
el)) - de(ej, ej))\u ,
- e~\Lk(&ik) ~
= 0.
But a : T
M
is an isomorphism, so [eit ej] =0
?
Theorem 3.3 (a) Let (M, o, e) be a massive F-manifold.
A vector field E is an Euler field of weight с eC if and only if
C.3)
(b)Let(M, p)= ni=iWb p) be the decomposition ofthe germ of a massive
F-manifold into irreducible germs of F-manifolds (Mk, o, ek).
(i) The space of (germs of) Euler fields of weight Ofor (M, p) is the abelian
Lie algebra ?;t=, С • ek.

26
Massive F-manifolds and Lagrange maps
(ii) There is a unique continuous function F : (L, ж '(/>)) —*• (С, 0) on the
multigerm (L, n~x(p)) which has value 0 on it~](p), is holomorphic on Lreg
and satisfies (dF)\Lreg = a\Lreg.
(Hi) An Eulerfield of weight с ф 0for(M, p) exists if'and onlyif"thisfunction
F is holomorphic. In that case, с ¦ a~](F) is an Euler field of weight с and
С • a~](F) + Yl'k=i"-" ' ek й the Lie algebra of all Euler fields on the germ
(M, p).
Proof. (a)Itis sufficientto prove this locallyforagerm(M, р)шй\ p e M—IC.
This germ is isomorphic to A". A vector field E = YL"i=\ е<еь 8i 6 Ом,Р, is an
Euler field of weight с if and only if ds,- = с ¦ d«,- (Theorem 2.11 and Example
2.12 (ii)).
Going into the proof of 3.2 (ii) =>• (iii), one sees that this is equivalent to C.3).
(b) The multigerm (L, тт~[(р)) has / components, and the space of locally
constant functions on it has dimension I. The function (multigerm) F exists
because a\Lreg is closed. This will be explained in the next section (Lemma
3.4). All statements follow now with (a). ?
3.2 Existence of Euler fields
By Theorem 3.2, the analytic spectrum (L, A) of an irreducible germ (M, p)
of a massive F-manifold is a germ of an (often singular) Lagrange variety, and
(L, A) '-*¦ (T*M, X) —у (М, p) is a germ of a Lagrange map. The paper [Gi2]
of Givental is devoted to such objects. It contains implicitly many results on
massive F-manifolds. It will be extremely useful and often cited in the following.
The question when does a germ of a massive F-manifold have an Euler field
of weight 1 is reduced by Theorem 3.3 (b)(iii) to the question when is the
function germ F holomorphic. Partial answers are given in Corollary 3.8 and
Lemma 3.9. We start with a more general situation, as in [Gi2, chapter 1.1].
Let(L,0) С {CN ,G) be a reduced complex space germ. Statements on germs
will often be formulated using representatives, but they are welldefined for the
germs, e.g. '<*\Lng is closed' fora 6 QkCN 0.
Lemma 3.4 Let a € ?2<J,,, 0 be closed on Lreg. Then there exists a unique
function germ F : (L, 0) —> (C, 0) which is holomorphic on LKg, continuous
on L and satisfies
Proof: The germ (L, 0) is homeomorphic to a cone as it admits a Whitney
stratification. One can integrate a along paths corresponding to such a cone
3.2 Existence of Euler fields
27
structure, starting from 0. One obtains a continuous function F on L, which is
holomorphic on Lreg because of da\Lrcg = 0 and which satisfies dF\L =a\L .
The unicity of F with value F@) = 0 is clear. ?
Which germs (L, 0) have the property that all such function germs are holo-
holomorphic on (L, 0)? This property has not been studied much. It can be seen
to be in line with the normality and maximality of complex structures and is
weaker than maximality.
It can be rephrased as Hqjv((L, 0)) = 0. Here #glV((L, 0)) is the cohomology
of the de Rham complex
(?$„,„/{<» 6 nj.Wi0 | w\Lns = 0}, d), C.4)
which is considered in [Gi2, chapter 1.1]. We state some known results on this
cohomology.
Theorem 3.5 (a) (Poincare-Lemma, [Gi2, chapter 1.1]) If (L, 0) is weighted
homogeneous with positive weights then Hqjv((L, 0)) = 0.
(b) ([Va5]) Suppose that (L, 0) is a germ of a hypersurface with an isolated
singularity, (L, 0) = (/"'(О), 0) С (C"+1, 0) and f : (C"+1, 0) -> (C, 0) is a
holomorphic function with an isolated singularity. Then
C.5)
-i- -dimOc,,,0/ f,^~
(с) (essentially Varchenko and Givental, [Gi2, chapter 1.2]) Let (L, 0) be
as in (b) with ц - x # 0. The class [q] e H?iv((L, 0)) ofrj e ^?«+',0 '¦* not
vanishing ifdrj is a volume form, i.e. dr) = hdxQ... dxn with h@) ф 0.
Remarks 3.6 (i) The proofs of (b) and (c) use the GauB-Manin connection for
isolated hypersurface singularities.
(ii) Theorem 3.5 (c) was formulated in [Gi2, chapter 1.2] only for n = l.The
missing piece of the proof for all n was the following fact, which at that time
was only known for n = 1: The exponent of a form hdxo ... dxn is the minimal
exponent if and only ifh(O) ф 0.
This fact has been established by M. Saito [SM4, C.11)] for all n.
(iii) By a result of K. Saito [SKI], an isolated hypersurface singularity
(L, 0) = (/~'@), 0) С (C"+1, 0) is weighted homogeneous (with positive
weights) if arid only if д — r = 0.
(iv) For us only the case n = 1 in Theorem 3.5 (b) and (c) is relevant.
Proposition 3.7, which is also due to Givental, implies the following:

28
Massive F-manifolds and Lagrange maps
Of all isolated hypersurface singularities (Z,, 0) = (/"'@), 0) С (C+1, 0)
only the curve singularities (n = 1) turn up as germs of Lagrange varieties.
These are, of course, germs of Lagrange varieties with respect to any volume
form on (C2, 0).
(v) If (L, 0) С ((S, 0), со) is the germ of a Lagrange variety in a symplectic
space S with symplectic form со, then the class [a] 6 H^iv((L, 0» of some a
with da = со is independent of the choice of a. It is called the characteristic
class of (L, 0) с ((S, 0), <и).
(vi) Givental made the conjecture [Gi2, chapter 1.2]: Let (Z,, 0) be an n-
dimensional Lagrange germ. IfH?iv((L, 0)) / 0 then H^iv{{L, 0)) # 0 and the
characteristic class [a] e Hq!v{{L, 0)) is nonzero.
It is true for n = 1 because of Theorem 3.5 and Remark 3.6 (iii). Givental sees
the conjecture to be analogous to a conjecture of Arnold's which was proved
by Gromov 1985 (cf. [Gi2, chapter 1.2]): any real closed Lagrange manifold
L с T*W has a nonvanishing characteristic class [a] e Я'(?> R).
Proposition 3.7 ([Gi2, chapter 1.1]) An n-dimensional germ (Z.,0) of a
Lagrange variety with embedding dimension embdim (Z,, 0) = n + к is a
product of a k-dimensional Lagrange germ (Z/, 0) with embdim (Z/, 0) = 2k
and a smooth (n — k)-dimensional Lagrange germ (L", 0); here the decompo-
decomposition of(L, 0) corresponds to a decomposition
(E, 0), со) = (E', 0), со') x (E", 0), со")
of the symplectic space germ (S, 0) Э (L, 0).
C.6)
Proof. If к < n then a holomorphic function f on S exists with smooth fibre
/~'@) Э L. The Hamilton flow of this function / respects L and the fibres
of /. The spaces of orbits in /~'@) and L give a symplectic space germ of
dimension 2n —2 and in it a Lagrange germ (e.g. [AGV1, 18.2]).
To obtain a decomposition as in C.6) one chooses a germ (E, 0) С E, 0) of
а 2и — 1-dimensional submanifold E in S which is transversal to the Hamilton
field H/ of /. There is a unique section v in (rS)^ with co(Hf, v) = 1 and
аАТрЪ, v) = 0 for p e E. The shift If of v with the Hamilton flow of /
forms together with H/ a 2-dimensional integrable distribution on S, because
This distribution is everywhere complementary and orthogonal to the inte-
integrable distribution whose integral manifolds are the intersections of the fibres
of / with the shifts of E by the Hamilton flow of /. This yields a decomposition
(S, 0) = (C2, 0) x (E П /~'@), 0). One can check that the symplectic form
decomposes as required. If к < п — 1 one repeats this process. D
3.3 Lyashko-Looijenga maps and graphs of Lagrange maps 29
Corollary 3.8 (a) Let {L, 0) be an n-dimensional Lagrange germ isomorphic
to (Z/, 0) x (C"~', 0) as complex space germ. Then (Z/, 0) is a plane cwve
singularity. The characteristic class [a] 6 Hgiv((L,Q)) is vanishing if and only
if{L', 0) is weighted homogeneous.
(b) Let (L,X) С (Т*М, X) be the analytic spectrum of an irreducible germ
(M, p) of a massive F-manifold. Suppose (L, X) = (Z/, 0) x (C", 0). Then
there exists an Eulerfield of weight 1 on (M, p) if and only if(L', 0) is weighted
homogeneous.
Proof, (a) Proposition 3.7, Theorem 3.5, and Remark 3.6 (iii).
(b) Part (a) and Theorem 3.3 (b)(iii).
?
In Proposition 5.27 for any plane curve singularity (Z/, 0) irreducible germs
of F-manifolds with analytic spectrum (Z,, X) = (Z/, 0) x (C"~l, 0) for some n
will be constructed. So, often there exists no Euler field of weight 1 on a germ
of a massive F-manifold. On the other hand, the Poincare-Lemma 3.5 (a) and
Proposition 3.7 say that an Euler field of weight 1 exists on a germ of a massive
F-manifold (M, p) if the multigerm (Z,, л ~' (/?)) of the analytic spectrum is at
all points of n~l(p) a product of a smooth germ and a germ which is weighted
homogeneous with positive weights. Also, we have the following.
Lemma 3.9 Let M be a massive F-manifold and F : L -> С a continuous
function with dF\Lreg = a\Lreg. Then a~l(F\(L - п~\1С))) is an Eulerfieldof
weight 1 on M — 1С. It extends to an Euler field on M if(L, X) is at all points
X € L outside of a subset of codimension > 2 a product of a smooth germ and
a germ which is weighted homogeneous with positive weights.
Proof. Suppose К С L is a subset of codimension > 2 with this property.
Then F is holomorphic in L - К because of the Poincare-Lemma 3.5 (a)
and Proposition 3.7. The Euler field extends to M - я(К). But jt(K) also
has codimension > 2. So the Euler field extends to M (and F is holomorphic
on L). О
3.3 Lyashko-Looijenga maps and graphs of Lagrange maps
In this section classical facts on Lagrange maps are presented, close to [Gi2,
chapter 1.3], but slightly more general. They will be used in sections 3.4-4.1.
LetZ, с Т*М be a Lagrange variety (not necessarily smooth) in the cotangent
bundle of an /n-dimensional connected manifold M. We assume:

30
Massive F-manifolds and Lagrange maps
(a) The projection ж : L —>• M is abranched covering of degree n, that is, there
exists a subvariety tfcAf suchthatrc : L-n~\K) -> М- К is a covering
of degree n (ж : L —> M is not necessarily flat).
(b) There exists a. generating function F : L —*¦ C, that is, a continuous func-
function which is holomorphic on Lng with &F\L = a\L (locally such a
function exists by Lemma 3.4).
Such a function F will be fixed. It can be considered as a multivalued function
on M - A"; the 1-graph of this multivalued function is L - n~[(K).
The Lyashko-Looijenga map A = (Л|,..., Л„) : M -»¦ С" of L С Т*М
and F is defined as follows: for q e M - K, the roots of the unitary polynomial
z" + 5Z"=i л;(?)г"~' are the values of F on n~\q). It extends to a holomorphic
map on M because F is holomorphic on Lreg and continuous on L.
The reduced Lyashko-Looijenga map A(red) = (A(*d) Л^'°) : M -»•
С of L С Г*М and F is defined as follows: for q e M -V, the roots
{fed)
=2 A{fed){q)zn~' are the values of F on
,(я) = ± Z!Xejr-i(?) F(A.). It also extends
of the unitary polynomial z" +
n~\q), shifted by their centre -± ?)
to a holomorphic map on M. Its significance will be discussed after
Remarks 3.11.
The front <!>?. of L С T*M and F is the image Im(F, pr) с С х М of
(F, pr) : L -> С x M. It is the zero set of the polynomial z" + YTi=\ л< '
z"~'. So, it is an analytic hypersurface even if F is not holomorphic on all
ofL.
Following Teissier ([Te2, 2.4, 5.5], [Lo2, 4.C]), the development ф2 с
РГ*(С х М) of this hypersurface Ot in С х М is defined as the closure
in PT*(C x M) of the set of tangent hyperplanes at the smooth points of Ф/..
It is an analytic subvariety and a Legendre variety with respect to the canonical
contact structure on РГ*(С х М).
The map
С x T*M -» P7"(C x M), (c,A.)i-> ((dz-A.)-'@),(c, p)) C.7)
(A. € г;м and dz - A. € 7-{*>p)(C x M) S ГС*С х г;М) identifies СхГМ
with the open subset in РГ*(С х М) of hyperplanes in the tangent spaces which
do not contain С ¦ ^. The induced contact structure on С х T*M is given by
the 1-form dz — a.
The following fact is well known. It is one way in which the relation between
Lagrange and Legendre maps can be made explicit (e.g. [AGV1, ch. 18-20]).
To check it, one has to consider F as a multivalued function on M - К and Ф/,
as its graph.
3.3 Lyashko-Looijenga maps and graphs of Lagrange maps 31
Proposition 3.10 The embedding С х Т*М ^- РГ*(С х М) identifies the
graphlm(F, id) С С x T*M of F : L ->• С with the development Ф^ of the
front Ф?.
Remarks 3.11 It has some nontrivial consequences.
(i) The development Ф^ is contained in the open subset of P7*(C x M) of
hyperplanes in the tangent spaces which do not contain ^. Therefore the vector
field -fz is everywhere transversal to the front Ф^..
(ii) The polynomial z"+5Z"=i л< z"~'has no multiple factors and the branched
covering Фх. -»¦ M has degree n: over any point p € M — К the varieties L
and Фь have n points and the points of Ф^ have n tangent planes; so, also Ф?,
has n points over a generic point p e M — K.
(iii) The graph Im(F, id) = Ф?. is an analytic variety even if F is not holo-
holomorphic on all of L.
(iv) The composition of maps Ф^ —-*¦ Im(F, id) —> L is a bijective mor-
phism. It is an isomorphism if and only if F is holomorphic on L. Also, the
continuous map L —> Ф^ is a morphism if and only if F is holomorphic on L.
(v) The Lagrange variety L <ZT*M together with the values of F at one point
of each connectivity component of L and any of the following data determine
each otheruniquely: the front Ф?., the development Ф?,, the Lyashko-Looijenga
map Л, the generating function F as a multivalued function on the base M.
To motivate the reduced Lyashko-Looijenga map, we have to talk about
Lagrange maps and their isomorphisms ([AGV1, ch. 18], [Gi2, 3.1]).
A Lagrange map is a diagram L '->¦ (S, со) —*¦ M where L is a Lagrange
variety in a symplectic manifold E, со) and S -> M is a Lagrange fibration. An
isomorphism between two Lagrange maps is given by an isomorphism of the
Lagrange fibrations which maps one Lagrange variety to the other.
An automorphism of T*M -> M as a Lagrange fibration which fixes the base
is given by a shift in the fibres,
T*M^T*M, Xh+X + dS,
where S : M -> С is holomorphic ([AGV1, 18.5]). So, regarding T*M -> M
as a Lagrange fibration means to forget the 0-section and the 1-form a, but to
keep the Lagrange fibration and the class a + {dS | S : M -> С holomorphic}
of 1-forms.
Corollary 3.12 Let L c-> T*M —>¦ M be as above (satisfying the assumptions
(a) and (b)) with I connectivity components and points X\, ... ,Xi, one in each
connectivity component. The data in (i)-(iii) are equivalent.

32
Massive F-manifolds and Lagrange maps
3.4 Miniversal Lagrange maps and F-manifolds
33
(i) The diagram Lc-^-T*M^-Masa Lagrange map and the differences
F(ki) - F(kj) e С of values of F,
(ii) the generating function F modulo addition of a function on the base,
(Hi) the reduced Lyashko-Looijenga map A(reJ) : M -*¦ С".
Proof. (i)=> (ii): Integrating the l-formsina+{d5 | S : M -*¦ С holomorphic}
gives (ii).
(ii) =4- (iii) Definition of A(red).
(iii) => (i): The map @, Л(гп/)) = @, A2red\ ..., A<rerf>) : M -»• C" is the
Lyashko-Looijenga map of a Lagrange variety in T*M which differs from the
original Lagrange variety only by the shift of d(^-Ai) in the fibres. The map
@, A(rerf)) determines this Lagrange variety and a generating function for it
because of Remark 3.11 (v). Then (i) is obtained. ?
3.4 Miniversal Lagrange maps and F-manifolds
The notion of a miniversal germ of a Lagrange map is central in Givental's
paper [Gi2]. We need a slight generalization to multigerms, taking a semilocal
viewpoint.
Let LcTMbeaLagrange variety with finite projection ж : L -*¦ M. The
germ at the base point p e M of L =->• T*M -*¦ M is the diagram
(L, jt-'Cp)) «-»¦ (T*M, T*pM) -> (M, p).
Here (L, л-'(р)) is a multigerm. The map of germs (T*M, T*M) -»• (M, p)
is the cotangent bundle of the germ (M, p); it is a germ in the base, but not in
the fibre.
For this diagram the morphisms
O
M,p
and
С 0
(c, X) i-
-> 3t.(Ol)p. (c. X) н> (с + e(X))|L
„М -> x*(OL)p/mp ¦ ir.(OL)p,
(c +a(X))|7r*(OL)p/m;, • jr,@L)p
C.8)
C.9)
are welldefined. Here X is in both cases a lift of X to 7 *M. These morphisms
are not invariants of the diagram as a germ at the base point p e M of a
Lagrange map because the identification of the Lagrange fibration with the
cotangent bundle of (M, p) is unique only up to shifts in the fibres and only the
class of 1-forms a + [dS \ S : M -*¦ С holomorphic} is uniquely determined
(cf. section 3.3).
But being an isomorphism or epimorphism in C.8) and C.9) is clearly a
property of the germ at p of the Lagrange map.
Definition 3.13 The germ atpeMofi^ T*M ->• M as a Lagrange
map is called miniversal (versal) if the morphism in C.9) is an isomorphism
(epimorphism) (cf. [Gi2, chapter 1.3]).
We are only interested in the case of a flat projection ж : L -*¦ M.Wellknown
criteria of flatness for finite maps (cf. [Fi, 3.13]) give the next lemma.
Lemma 3.14 The following conditions are equivalent.
(i) Thegermat p e MofL =->• T*M -*¦ M as a Lagrange map is miniversal
with flat projection ж : (L, n~x(p)) -*¦ (M, p),
(ii) it is miniversal with deg ж = 1 + dim M,
(iii) the morphism in C.8) is an isomorphism,
(iv) the Lagrange map is miniversal at all points in a neighbourhood of
p&M.
Example 3.15 A miniversal germ at a base point of a Lagrange map with a
projection ж : L -»• M which is not flat is given by the germ at 0 e C2 of the
Lagrange fibration C4 ->• С2, (уг.уз.Ь^з) •->¦ (t2, b) with со = dy2dt2 +
dy^db and by the Lagrange variety L which is the union of two appropriate
planes and which is defined by the ideal
(Л. Уз) n(y2-t2,y3- h) = (y2, уъ) ¦ (Уг -h,y3- h). C.10)
Now let M be a massive n-dimensional F-manifold with analytic spectrum
L С T*M. Then (L, n~\p)) ^ (T*M, T*M) -+ (M, p) is for any p e M a
versal, but not a miniversal germ at the base point peMofa Lagrange map.
But there is a miniversal one.
The germ of the fibration at p whose fibres are the orbits of the unit field
e is denoted by pre : (M, p) -*¦ (M(r), pw). The fibrewise linear function on
T*M which corresponds to e is called y\. Its Hamilton field ? := Hyi is a
lift of e to T*M. It leaves the hypersurface ^,"'A) С Т*М and the Lagrange
variety L С yf 'A) invariant. The orbits of ~e in yj"'(l) form a germ of a
2n — 2-dimensional symplectic manifold with a Lagrange fibration, which can
be identified with the cotangent bundle

34
Massive F-manifolds and Lagrange maps
3.5 Lyashko-Looijenga map of an F-manifold
35
But this identification is only unique up to shifts in the fibres. The orbits of 7
in L form a Lagrange variety L(r) С T*Mir).
The germ at p(r) e Mir) of the diagram L(r) <^-> T*M^r) -> M(r) is unique
up to isomorphism of germs in the base of Lagrange maps. It will be called the
restricted Lagrange map of the germ (M, p) of the F-manifold M. An explicit
description will be given in the proof of the next result.
Theorem 3.16 (a) The restricted Lagrange map of the germ(M, p) of a massive
F-manifold is miniversal with flat projection n^ : L(r) -> Л/(г).
(b) It determines the germ (M, p) of the F-manifold uniquely.
(c) Any miniversal germ at a base point of a Lagrange map L' =-> T*M' ->
M' with flat projection L' —*¦ M' is the restricted Lagrange map of a germ of a
massive F-manifold.
Proof: (a) In order to be as explicit as possible we choose coordinates t —
(t\,t') = (t\,...,tn):(M,p)^ (C", 0) with e(t\) = 1. The dual coordinates
on T*M are (yi,..., у„) = (уь у') = у. The multiplication is given by ^ о
;?- = J^k afj(t')~ and the analytic spectrum L is
I ? \(y,t) e C" x (C,0) | y, = 1, yiyj = ?4(r')yJ. C.11)
The restricted Lagrange map is represented by the Lagrange fibration
С x (C-'.O)-* (C"-',0), (y',t')h+ t' C.12)
with canonical 1-form a' := ?/>2 У'1^' ап<^ by the Lagrange variety
{(/, t') 6
1,0) | yiyj
j = 4(г') + ^<а')Л for i, j > 2J = L(r).
<fc>2 J
C.13)
The equations for the Lagrange variety in C.13) show that the morphism in
C.8) for this Lagrange map with fixed canonical 1-form a' is an isomorphism.
This implies that the restricted Lagrange map for (Af, p) is miniversal with flat
projection.
(b) and (c) Any miniversal germ at a base point p' e M' of a Lagrange map
L' °-> T*M' —*¦ M' with flat projection л' : L' —*¦ M' can be represented by a
Lagrange fibration as in C.12) and a Lagrange variety as in C.13).
Defining L by C.11) and M := С x (M', p') and e := ^-, one obtains an
F-manifold with unit field e and analytic spectrum L.
It remains to show that this does not depend on the way in which the Lagrange
fibration is identified with the cotangent bundle of (C"""!, 0) in C.12). But one
sees easily that a shift in the fibres of C.12) of the type y,- i-> y,- + ff for some
holomorphic function 5 : (C"',0) -> С on the base corresponds only to a
change of the coordinate fields ^-,..., ^- and the coordinate t\ in M and thus
to a shift of the section {0} x (M\ p') in M -> (M1, p'). It does not affect L
and the multiplication on (M, p). D
Let (M, p) be a germ of a massive F-manifold. The germ
дрю := [X 6 TM,p I [e, X] = 0}
C.14)
is a free 0WM]/7ir)-module of rank n. It is an Ом"\р^-algebra because of
Liee(o) = 0 • o. The functions a(X) for X e Qpm are invariant with respect to
? and induce holomorphic functions on L(r). One obtains a map
Lemma 3.17 The map a(r) is an isomorphism of ОMi,i ^-algebras.
C.15)
Proof. The isomorphism a: Тм,р -*¦ {ix^Odp maps the e-invariant vector fields
in (M, p) to the (T-invariant functions in GГ*0/.)р. ?
This isomorphism a(r) is closely related to C.8) for the restricted Lagrange
map of (M, p):
An isomorphism as in C.8) requires the choice of a 1-form for its Lagrange
fibration. The choice of a function t\ : (M, p) -» (C, 0) with e(t\) = 1 yields
such a 1-form: the 1-form which is induced by a — dt\ (a — dt\ on T*M is
«^-invariant and vanishes on "e and induces a 1-form on the space of «Г-orbits of
The choice of t\ also yields an isomorphism
ОМ(-),ри Ф TM(o,pM -> CV'>,,<" • e Ф {X e ?PM I X(ti) = 0} = дрю. C.16)
One sees with the proof of Theorem 3.16 (a) that the composition of this iso-
isomorphism with a(r) gives the isomorphism in C.8) for the restricted Lagrange
map of (M, p) (the germ {M, p) in C.8) in this case is (M(r), p(r))).
3.5 Lyashko-Looijenga map of an F-manifold
Definition 3.18. Let (M, o, e) be a massive n-dimensional F-manifold with
analytic spectrum L С Т*М.

36
Massive F-manifolds and Lagrange maps
(a) A generating function F for (M, o, e) is a generating function for L,
that is, a continuous function F : L —> С which is holomorphic on Lreg with
dF|tl4 =«!*„.
(b) Let F be a generating function for (M, o, e).
(i) The bifurcation diagram В с M of(M,o,e, F)isthe set of points p e M
such that F has less than л different values on я (p).
(ii) TheLyashko-Looijengamap A = (Ль ..., Л„): M -*¦ С oi(M, о, е, F)
is the Lyashko-Looijenga map of F as the generating function for L С
T*M (cf. section 3.3).
(iii) The discriminant V С M of (А/, о, г, F) is V := Л~'@).
The discriminant will be discussed in section 4.1. A generating function for
an F-manifold exists locally (Lemma 3.4), but not necessarily globally.
A holomorphic generating function F corresponds to an Euler field E :=
a~l(F) of weight 1 (Theorem 3.3); then the values of F on л~1(р), р е М,
are the eigenvalues of Eo : TpM —*¦ TpM. The objects B, A, V of Definition
3.18 (b) are welldefined for (M, o, e, E) if E is such an Euler field.
The restriction of л : L —>¦ M to the complement of the caustic /C is a covering
7Г : n~[(M — /C) -*¦ M -K. of degree n, and ж'1 (М -К,) is smooth. Hence a
generating function F is holomorphic on ж~1(М — /C) and corresponds to an
Euler field E on M — /C. Results and examples about the extendability of E to
M are given in Lemma 3.9 and Theorem 5.30.
The bifurcation diagram В of (M, o, e, F) contains the caustic /C. The caustic
is a hypersurface or empty (Proposition 2.6) and invariant with respect to the
unit field e (Remark 2.9 (vii)). The bifurcation diagram has the same properties:
the restriction of F to an open set U С M — К with canonical coordinates
(mj ,..., «„) corresponds to an Euler field E = ?(«, + ri)ei f°r some r,- 6 C,
and the bifurcation diagram is the hypersurface
В П U = U П [и | и,- + r,- = My + ry for some i # /}.
It is invariant with respect to e because of е(и,- — м;) = 0.
The Lyashko-Looijenga map for the F-manifold A" = (С", о, e) (Example
2.12 (ii)) with Euler field E = Y, «;«•; and Euler field-function F := a(E) is
л(«): С" -> С", « h> ((-1L-("));
C.17)
here o\ (и), ...,ап{и) are the symmetric polynomials. The group of automor-
automorphisms of the F-manifold A" which respect the Euler field E is the sym-
symmetric group Sn which permutes the coordinates u\,... ,и„. The map Л(п)
3.5 Lyashko-Looijenga map of an F-manifold
37
is the quotient map for this group. It is branched along the bifurcation diagram
В = [и | и; = uj for some i # j). The image Л(л)(#) is the hypersurface
Vм := la e C" | z" + J] a;z"~' has multiple roots} С С". C.18)
The restriction Л(п) : С" — В ->¦ С - Vм induces an F-manifold structure on
?n _ ?>«, with unit field
e(n) ._ dA<«)(e) _ _„
and Euler field
K-17- C19)
oat
C.20)
This F-manifold (C - ?>(n), o, e(n)) will be denoted by A\/Sn.
Theorem 3.19 Let (M, o,e)bea massive F-manifold with generating function
F : L —*¦ С and Lyashko-Looijenga map A : M —>¦ C.
77ien Л-'ф(п)) = В anddA(e) = e(n). Г/ie restriction A : M - В ->¦ С -
P(n) и аи immersion and locally an isomorphism of F-manifolds. It maps the
Euler field a1 (F\M-b) on M - В to the Euler field Ew.
Proof In M — /C the multiplication is semisimple and locally the values of the
generating function are canonical coordinates. The map Л factors on M — В
locally into an isomorphism to A" and into the map Л(п). ?
The most important part of Theorem 3.19 is that A : M - В-*¦ С - V(n) is
locally biholomorphic. The following statements for germs will also be useful.
Lemma 3.20 Let (M, p) = I~L=i(^b P) be я germ of a massive F-manifold
with analytic spectrum Land with decomposition into irreducible germs (M^, p)
of dimension n^, Ylnk — n-
(a) There exists precisely one generating function on the multigerm
(L, 7t~l(p))forany choice of its values on 7r~'(p) = {k\,..., Л;}.
(b) Choose a function t\ : (M, p) ->¦ С with e(t\) = 1.
The values of a generating function for the points in L above an orbit of e
are of the form t\ +a constant.
The entry Aj of a Lyashko-Looijenga map A : (M, p) —*¦ С is a polynomial
of degree i in t\ with coefficients in {g e Ом p \ e(g) = 0} andleading coefficient
(-1УС).

38
Massive F-manifolds and Lagrange maps
(c) Choose representatives Mi for the germs (Mk, p)and Lyashko—Looijenga
maps A[k] : Mk -»¦ C". Then the function A = (A,,..., Л„): П* Mk -> С
which is defined by
1=1
,?-' = f] [z"
C.21)
is a Lyashko-Looijenga map for the representative fj Mk of the germ (M, p).
Any Lyashko—Looijenga map for (M, p) is of this type.
Proof, (a) Lemma 3.4.
(b) It suffices to prove the first part for an orbit of e in M — JC. There the
generating function comes from an Euler field. The formulas B.9) and B.10)
imply Liee(?o) = id. The values of F are the eigenvalues of Eo.
(c) The map A[k] corresponds to an Euler field ?№1 (at least) on Mk - BMt.
The sum J2k ?W is an Euler field on TlWk - BMk) by Proposition 2.10. The
corresponding generating function extends to f] Mk and has the given Л as its
Lyashko-Looijenga map. The last statement follows with (a). D
Consider the projection pre : (M, p) —*¦ (M(r\ p^) whose fibres are the
orbits of e (section 3.4). The e-invariant hypersurfaces В and /C project to
hypersurfaces in M^r\ which are called the restricted bifurcation diagram 23(r)
and the restricted caustic /C(r). In section 3.3 the restricted Lagrange map was
defined as the germ at p(r> e M(r) of a Lagrange map L(r) ^ T*M(r) -»¦ M(r\
Because of Corollary 3.12 the notion of a reduced Lyashko-Looijenga map is
welldefined for the restricted Lagrange map (independently of the identification
of the Lagrange fibration with the cotangent bundle of M(r)).
The space of orbits of the field eM (formula C.19)) in C" can be identified
with (a e C" | fli = 0) = {0} x С = С" and is equipped with the co-
coordinate system (fl2,..., an) = a'. The projection to this orbit space is denoted
by pr(n) : С -»¦ С1, and the image of ?>(n) is
_,):= l
чл-l
i=2
has multiple roots |
C.22)
(it is isomorphic to the discriminant of the singularity or F-manifold An-\,
cf. section 5.1).
Corollary 3.21 Let (M, p)be the germ of a massive F-manifold, F a generating
function, A (ASndr>) the (reduced) Lyashko-Looijenga map of(M, o, e, F).
The map A^^ : (M,p) -*¦ C is constant along the orbits of e. The
induced map A(rerfXl") : (M(r\ p(r)) -*¦ C" is a reduced Lyashko-Looijenga
3.5 Lyashko-Looijenga map of an F-manifold
39
ap for the restricted Lagrange map. The following diagram commutes, the
'diagonal morphism is Л"*" = pr<"> о Л = Л"*"™ о pre,
т>
(М,р)
С"
(M{r\ Pir)) > С'
1И-1
C.23)
The restriction
is locally biholomorphic.
- X>(An_i)
C.24)
Proof. The map A(red) is constant along the orbits of e because of Lemma
3.20 (b). The formulas C.11), C.12), C.13) show that A<re</)(r) is a reduced
Lyashko-Looijenga map for the restricted Lagrange map. The rest follows from
Theorem 3.19. П

Chapter 4
Discriminants and modality of F-manifolds
4.1 Discriminant of an F-manifold
41
Discriminants play a central role in singularity theory. Usually they have a rich
geometry and say a lot about the mappings or other objects from which they
are derived. The discriminant V of a massive F-manifold M with a generating
function (cf. Definition 3.18) is an excellent model case of such discriminants,
having many typical properties.
Together with the unit field it determines the whole F-manifold in a nice
geometric way. This is discussed in section 4.1 (cf. Corollary 4.6). In section
4.3 results from singularity theory are adapted to show that the discriminant
and also the bifurcation diagram are free divisors under certain hypotheses.
The classification of germs of 2-dimensional massive F-manifolds is nice
and is carried out in section 4.2. Already for 3-dimensional germs it is vast
(cf. section 5.5). In section 4.4 the Lyashko-Looijenga map is used to prove
that the automorphism group of a germ of a massive F-manifold is finite. There
also the notions modality and /^-constant stratum from singularity theory are
adapted to F-manifolds. In section 4.5 the relation between analytic spectrum
and multiplication is generalized. This allows F-manifolds to be found in natural
geometric situations (e.g. hypersurface and boundary singularities) and to be
written down in an economic way (e.g. in 5.22, 5.27, 5.30, 5.32).
4.1 Discriminant of an F-manifold
Let (M, o, e, F) be a massive n-dimensional F-manifold with a generating
function F : L -> С and Lyashko-Looijenga map Л = (ЛЬ...,Л„):М-»
С"; the discriminant of (M, o, e, F) is the hypersurface V = Л~'@) с М
(Definition 3.18).
If F is holomorphic and E = a~'(F) is its Euler field then Л„ = (-1)" •
det(F,o), and the discriminant is the set of points where the multiplication with
E is not invertible.
By the definition of Ли, the discriminant is V = n(F ' @))/Theorem 4.1 will
give an isomorphism between F @) and the development V С VT*M of V.
We need an identification of subsets of T*M and FT*M. The fibrewise linear
function on T*M which corresponds to e is called y\. The canonical map
3>j~'(l)—у ?Т*М D.1)
identifies yf'A) С Т*М with the open subset in VT*M of hyperplanes in the
tangent spaces of M which do not contain С • e.
The restriction to yf'O) of the canonical 1-form a on T*M gives the contact
structure on з>Г'A) which is induced by the canonical contact structure on
Theorem 4.1 Let(M, o, e, F)andF~\G) С L С y^d) С Т* М be as above.
The canonical map y^1 A) <-» PT*M identifies F~l@) with the development
V С РТ*М of the discriminant T>.
Proof. We want to make use of the discussion of fronts and graphs (section
3.3) and of the restricted Lagrange map (section 3.4). It is sufficient to consider
the germ (M, p) for some p 6 V. We choose coordinates (t\,..., tn) and
(yi, ¦ • •. Уп) as in the proof of Theorem 3.16. The generating function F for
I <^. j*M -> M takes the form
F(y', t) = h + F(r)(/, t'), ' D.2)
where Fw is a generating function of L(r) with respect to C.13), C.12), and
a' = Хл>2 yi&i- The isomorphism with a sign
(-«i. pre) :(«,?)->(Cx Mw, 0 x p(r>) D.3)
maps the discriminant to the front Im(FCr), я(г)) of Lw and F(r).
The development of this front is identified with the graph Im(F(r), id) С
С x T*Mir) of F(r> : L(r) -> C, by Proposition 3.10 and the embedding (cf.
formula C.7))
С x T*Mir) <-> РГ*(С x M(r)),
(-fi, X) ^ ((df, + X)-'(O), ( - tu ttw(X))). D.4)
But D.1), D.3), and D.4) together also yield an isomorphism yj~'A) -> С х
Г*М(Г>, which maps F~' @) to this graph. D
Remarks 4.2 (i) The most important consequence is that V С FT*M does not
contain a hyperplane which contains С • e (cf. Remark 3.11 (i)). Therefore the
unit field e is everywhere transversal to the discriminant V.
40

42
Discriminants and modality of F-manifolds
(ii) In the proof only the choice of t\ is essential. It is equivalent to various
other choices: the choice of a section of the projection (M, p) —*¦ (MM, p^r)),
the choice of a 1-form a' for the Lagrange fibration in the restricted Lagrange
map.
(iii) Fixing such a choice of t\, one obtains together with a' and F(r) a
Lyashko-Looijenga map A(r) : (M(r\ p{r)) -» С" for F{r). Then D.3) iden-
identifies the entry Л„ of the Lyashko-Looijenga map Л : (M, p) -*¦ C" with the
polynomial (-f,)" + Y!U A,(r)(-f,)"-'.
(iv) The set F~l@) С L is not an analytic hypersurface of L at points of
LSing where F is not holomorphic. But Theorem 4.1 shows that it is everywhere
a subvariety of L of pure codimension 1. Examples where F is not holomorphic
will be given in section 5.5.
(v)Let(M, p) = ni=i(^b p)bethedecompositionofagerm(M, p)intoir-
p)intoirreducible germs of F-manifolds. A Lyashko-Looijenga map Л : (M, p) -*¦ С
corresponds to Lyashko-Looijenga maps A'*' : {Mk, p) —*¦ C* for the irre-
irreducible germs, in a way which was described in Lemma 3.20 (c). One obtains
especially An = ]~It Aj*], and the germ (V, p) of the discriminant is
. P) = \J
<k
UMJ-p
D.5)
the union of products of smooth germs with the discriminants for the irreducible
germs (of course, it is possible that ((Aj*])~'(O), p) = 0 for some or all k).
(vi) The development V of the discriminant V gives the tangent hyperplanes
to V. Let (M, p) = Y\'k^i(Mk, p) be as in (iv), and л~\р) = {Xu ..., A,}.
Theorem4.1 says that the tangenthyperplanes to (?>, /?)arethosehyperplanes
A^'@) С ТрМ for which F(A*) = 0.
Especially, if/ = 1 and F(A.,) = 0, then Aj"'(O) С TPM is the nilpotent
subalgebra of ГРМ and the unique tangent hyperplane to (V, p). The general
case fits with Lemma 2.1 (iv) and D.5).
(vii)The equality a\Lirg = dF|Lres shows immediately that F~'@) С yf 'A)
is a Legendre subvariety.
(viii) If Л € F~'@) П T*M, there is a canonical projection from the Leg-
Legendre germ (JF~' @), X) to one component of the Lagrange multigerm (L(r\
n(r) (pM)). it is a bijective morphism. It is an isomorphism if and only if F
is holomorphic at (L, A.) (see Remark 3.11 (iv)).
One can recover the multiplication on a massive n-dimensional F-manifold
M from the unit field e and a discriminant V if the orbits of e are sufficiently
large. To make this precise, we introduce the following notion.
4.1 Discriminant of an F-manifold
43
Definition 4.3 A massive F-manifold (M, o, e, F) with generating function F
is in standard form if there exists globally a projection pre : M —>• M(r) to a
manifold M(" such that
(a) the fibres are the orbits of e (and thus connected),
ф) they are with their affine linear structure isomorphic to an open (connected)
subset of C,
(y) the projection pre : T> —> M^ is a branched covering of degree n.
Remarks 4.4 (i) If (M, o, e, F) is a massive F-manifold with generating func-
function F and properties (a) and (/3) then
1
( A,,prJ
V n )
M
С x M(r)
D.6)
is an embedding because of e{—\i^\) = 1 (Lemma 3.20 (b)).
The F-manifold M can be extended uniquely to an F-manifold isomorphic
to С x M(r). Also the generating function F can be extended. The discriminant
of this extended F-manifold satisfies (y) because of V = A~'@) and Lemma
3.20 (b).
(ii) For (M, o, e, F) as in (i) the coordinate t\ :— — ^A\ is distinguished
and, up to the addition of a constant, even independent of the choice of F.
Nevertheless it does not seem to have good properties: In the case of the simplest
3-dimensional irreducible germs of F-manifolds, A3, B3, Щ, it is not part of
the coordinate system of a nice normal form (section 5.3). Using the data in
[Du2] one can also check that — ^Ai is not a fiat coordinate of the Frobenius
manifolds A3, B3, Я3.
Corollary 4.5 Let (M, o, e, F) be a massive F-manifold with generating func-
function F and in standard form.
(a) The branch locus of the branched covering pre : V -*¦ M(r' is Vsing, the
set pre(DSins) of critical values is the restricted bifurcation diagram B^ =
pre(B).
(b) The union of the shifts of F~\0) with the Hamilton field 7 = Hyi is the
analytic spectrum L С Т*М.
(c) The data (M, o, e, F) and {M, e, V) are equivalent.
Proof, (a) Theorem 4.1 implies that all tangent hyperplanes to T> are transversal
to the unit field. Therefore the branch locus is only T>smg.
(b)and(c)ThediscriminantI?andtheunitfieldedetermineF~l(O) С yf'(l)
because of Theorem 4.1. The uni on of shifts of F ~' @) with the Hamilton field
"e = Hyi is finite of degree n over M because of (y) and it is contained in L,

44
Discriminants and modality of F-manifolds
so it is L. The function F : L -*¦ С is determined by F~'@) С L and by the
linearity of F along the orbits of e". ?
Theorem 4.1 and Corollary 4.5 (b) give one way to recover the multiplication
of a massive F-manifold (A/, o, e, F) in standard form from the discriminant T>
and the unit field e. The following is a more elementary way.
Corollary 4.6 Let (M, o, e, F) be a massive n-dimensional F-manifold with
generating function F and in standard form. The multiplication can be recover-
recovered from the discriminant T> and the unit field e in the following way. The multipli-
multiplication is semisimple outside of the bifurcation diagram В = pr~l(pre(Vsing)).
For a point p € M — B, the idempotent vectors е,{р) е TpM with е,{р) о
ejip) — Sjjeiip) are uniquely determined by (i) and (ii):
(i) the unit vector is e{p) = 2^"=1 ei(p),
(ii) the multigerm (P, T> П pr~\pre(p))) has exactly n tangent hyperplanes;
their shifts to TpM with e are the hyperplanes ф,^ С • e,- С ТрМ, к =
1.....П.
Proof. Remark 4.2 (vi).
D
4.2 2-dimensional F-manifolds
The only 1-dimensional germ of an F-manifold is A\ (Example 2.12 (i)). The
class of 3-dimensional germs of massive F-manifolds is already vast. Examples
and a partial classification will be given in section 5.5. But the classification of
2-dimensional germs of F-manifolds is nice.
Theorem 4.7 (a) The only germs of 2-dimensional massive F-manifolds are,
up to isomorphism, the germs him), m e N>2, with I2B) = A2, I2C) = A2,
/2D) = Въ /2E) =: H2, /2F) = G2,from Example 2.12 (iv):
The multiplication on (M, p) = (C2, 0) with coordinates t\, t2 and Si := ^
is given by e := <5| and S2 о S2 = t™~2 ¦ S\. An Euler field of weight 1 is
E = /,Si+ ^t2&2. Its discriminant is V = {t\t\-^t% — 0}. The germ I2(m)
of an F-manifold is irreducible for m > 3 with caustic and bifurcation diagram
/С = В = {t 112 = 0}. The space of Euler fields of weight d is d ¦ E + С • e for
m>3.
(b) The only germ of a 2-dimensional not massive F-manifold is the germ
(C2, 0) from Example 2.12 (v) with multiplication given by e := S\ andS2o82 =
0. The caustic is empty. An Euler field of weight 1 is E = t[&\. The space of all
Euler fields of weight 0 is {e,5| + e2(t2)B2 \ e{ e C, e2(t2) e C{t2}}.
4.2 2-dimensional F-manifolds
45
Proof (a) Givental [Gi2, 1.3, p. 3253] classified the 1-dimensional miniversal
germs of Lagrange maps with flat projection. Together with Theorem 3.16 this
yields implicitly the classification of the 2-dimensional irreducible germs of
massive F-manifolds. But we can recover this in a simple way and we need to
be more explicit.
Let (M, p) be a 2-dimensional germ of a massive F-manifold with projection
pre : (M, p)->- (Af(r), p(r))to the space of orbits of e. There isaunique genera-
generating function F : (L, л~1(р)) -*¦ (С, 0). Its bifurcation diagram В с М and
restricted bifurcation diagram B(r) С A/(r)arethehypersurfacesS = pr~^(p<r^)
By Corollary 3.21, the reduced Lyashko-Looijenga map Л*""*^:
(M(r\ p(r)) —*¦ (C, 0) of the restricted Lagrange map is a cyclic branched cov-
covering of some order in, and the Lyashko-Looijenga map Л : M —*¦ C2 is also a
cyclic branched covering of order in, branched along B.
Because of Corollary 3.12 and Theorem 3.16, this branching order in of
д(г"/)(г) determines the germ {M, p) of the F-manifold up to isomorphism. It
remains to determine the allowed in and explicit formulas for the F-manifolds.
Now consider the manifolds C2 with multiplication on ГС2 given by e = Si
and S2oS2 = t™~2 ¦ h\. The analytic spectrum L С Т*С2 is
L - {(yi, y2,
D-7)
It is an exercise to see that L is a Lagrange variety with generating function
F = t\ + ^y2h with respect to a — y\dti+y2dt2, i.e. onehas(a— dF)\Lnsg = 0.
Then this gives an F-manifold and E = a~' (F) — t\S\ + ^t2b2 is an Euler field
of weight 1. The Lyashko-Looijenga map Л is
Л : С2 -+ С2, (г,, t2) t+(-2tut2- ^
(-
D.8)
It is branched along В = {t \ t2 = 0} of degree m.
So I2(m) is the desired F-manifold for any branching order m = m > 2. The
same calculation yields for m = m = 1 the F-manifold A1/S2 on C2 — {t 112 = 0}
(section 3.5) with meromorphic multiplication along {t \t2 = 0}.
(b) Let (M, p) = (C2, 0) be the germ of a 2-dimensional not massive F-
manifoldwithe — <5i.ThenLiee(o) = 0and[e, S2] = OimplyLiee(<52 0<52) — 0.
Hence <52 о <$2 = p{t2)8i + y(h)&2 for some 0(f2), yUi) e C{t2}.
The field S2 := S2 - {y(t2)Si satisfies S2 о S2 = (fi(t2) + \y(t2f)8{ and
[<$,, S2] = 0. Changing coordinates we may suppose S2 = S2, y(t2) = 0.
The analytic spectrum is L = {{y\, y2% t\, t2) | y\ = 1, y2 • y2 = fi{t2)}. The
F-manifold is not massive, hence fi(t2) = 0. One checks with B.7) and B.8)

46
Discriminants and modality of F-manifolds
easily that this multiplication gives an F-manifold and that the space of Euler
fields is as claimed. ?
Let us discuss the role which 2-dimensional germs of F-manifolds can play
for higher dimensional massive F-manifolds. The set
a € С | z" + ]Г) ац"~1 has a root of multiplicity > 3 J С Vм с С"
D.9)
is an algebraic subvariety of C" of codimension 2 (see the proof of Proposition
2.5). Given a massive F-manifold M, the space
/CC) := {p € M | P(TpM) >-C,l l)}ciCcM D.10)
of points p such that (M, p) does not decompose into 1- and 2-dimensional
germs of F-manifolds is empty or an analytic subvariety (Propostion 2.5).
Theorem 4.8 Let (M, o, e) be a massive F-manifold with generating func-
function F.
(a) The function F is holomorphic on л~х{М — /C<3)) and gives rise to an
Euler field of weight 1 on M — KP\
(b) //codim/CC) > 2 then F is holomorphic on L and E = a~\F) is an
Euler field of weight 1 on M.
(c) One has /CC) С A^"'3'), and A-'CD("'3)) -/C<3> is analytic of pure
codimension 2. Thus codim/CC) > 2 «=>¦ codimA^"'3') > 2.
(d) The restriction of the Lyashko—Looijenga map
A : M - A-'(D("-3)) -» C" -D("'3)
is locally a branched covering, branched along В — A (X>("'3)).
IfpeB- A-'fD'"'3') and A(p) e Х>(л) - D("-3) ore smootfi pomta o/fAe
hypersurfaces В andV^ and if there the branching order is m, then (M, p) is
the germ of an F-manifold of type h(m) x A"~2.
Proof: (a) and (b) Each germ (L, Xk) of the analytic spectrum (L, n~](p)) =
(L, {Ль ..., Л/}) of a reducible germ (M, p) = Ylk(Mk, P) is the product of a
smooth germ with the analytic spectrum of (M^, p). The analytic spectrum of
him) (m > 2) is isomorphic to (C, 0) x ({y2, t2) | y\ = t™^2}, 0). One applies
Lemma 3.9.
(c) A Lyashko-Looijenga map of h (m) is a cyclic branched covering of order
m, branched along the bifurcation diagram. This together with Lemma 3.20 (c)
4.3 Logarithmic vector fields
47
implies that locally around a point p € M — /CC) the fibres of the Lyashko-
Looijenga map A : M —*¦ C" are finite. Therefore codim/v/(A~'(I>('l-3)), p) =
codime(D(;l-3))=2.
(d) The map Л determines the multiplication of the F-manifold M (Theorem
3.19). One uses this, Lemma 3.20 (c) and properties of h(m). Q
Many interesting F-manifolds, e.g. those forhypersurface singularities, boun-
boundary singularities, finite Coxeter groups (sections 5.1, 5.2, 5.3), satisfy the
property codim/CC) > 2 and have an Euler field of weight 1.
4.3 Logarithmic vector fields
K. Saito [SK4] introduced the notions of logarithmic vector fields and free
divisors. Let H С M be a reduced hypersurface in an я-dimensional manifold
M. The sheaf Der^ (log H) С Тм oflogarithmic vector fields consists of those
holomorphic vector fields which are tangent to Hreg. This sheaf is discussed
in detail in section 8.1. There it is shown that it is a coherent and reflexive
Ом-niodule. The hypersurface H is a free divisor if Der^(log H) is a free
См-module of rank n.
The results in this section are not really new. They had been established
in various generality by Bruce [Bru], Givental [Gi2, chapter 1.4]), Lyashko
[Lyl][Ly3], K. Saito [SK6][SK9], Terao [Тег], and Zakalyukin [Za] as results
for hypersurface singularities, boundary singularities or miniversal Lagrange
maps. But the formulation using the multiplication of F-manifolds is especially
nice.
Theorem 4.9 Let(M, o, e) be a massive F-manifold with Euler field E of weight
1, generating function F — a(?) and discriminant V = (det(?o))~'@) =
(a) The discriminant is a free divisor with Der^(log V) = E о Тм-
(b) The kernel of the map
0) D-11)
is
ker nv = EoTM= DerM(log V).
D.12)
Proof, (a) The sheaf E о Тм is a free 0M-module of rank n. Therefore (a)
follows from D.12).

48
Discriminants and modality of F-manifolds
(b) The Од*-module я,С/г-1(о) has support V. Equation D.12) holds in
M - 23. The set V П В = Vsing (cf. Corollary 4.5 (a)) has codimension 2
in M. The following shows that it is sufficient to prove D.12) in V - Vsing.
Let M — Vsing <-+ M be the inclusion. The Riemann extension theorem says
OM — i*(OM-vlins) (cf. for example [Fi, 2.23]). Now E oTM satisfies
EoTM = /»(? о TM\M^t>siJ D.13)
because it is a free CM-module. The sheaves kerap and DerA/(log2?) satisfy
the analogous equations because of their definition. Hence D.12) holds in M if
it holds in V - Vsing.
Let p 6 V—Vsing. We choose a small neighbourhood U of p with canonical
coordinates u\,...,un centred at p, with РП U = {и | u{ =0} and with Euler
field E = и\в\ + ?,>2(«; + л>; for some г,- е С — {0}. With the notation of
the proof of Theorem 3.2 (ii) =Ф- (iii) we have a = J2 *,d«,-,
F~\0) П n~4U) = {(x, «) | xj = Su, щ = 0},
and for any vector field X = X! ?/*i € T^(U)
Therefore
= f,@, иг,..., «„)¦
i=2
= ? о TW)/, = DerWiP(log23).
D.14)
П
Remark 4.10 One can see Theorem 4.9 (a) in a different way: there is a criterion
of K. Saito [SK4, Lemma A.9)]. To apply it, one has to show
[E°TM,EoTM]cEoTM. D.15)
With B.5) and B.6) one calculates for any two (local) vector fields X, Y
[EoX,EoY] = Eo ([X, EoY]-[Y,E6X]-Eo[X, Y]). D.16)
In the rest of this section (M, о, е, Е) will be a massive F-manifold which
is equipped with an Euler field E of weight 1 and which is in standard form
(Definition 4.3). The map pre : M -*¦ M(r) is the projection to the space of
orbits of e. The sheaf of e-invariant vector fields
Q := {X € (preXTM | [e, X] = 0}
D.17)
49
4.3 Logarithmic vector fields
is a free O^o-module of rank n. Because of Liee(o) = 0 it is also an CVo-
algebra.
Theorem 4.11 Let (M, o, e, E) be a massive F-manifold with Euler field E of
weight 1 and in standard form,
(a)
(рге)*Тм = G® (pre)*{E о Тм).
(b) The kernel of the map
(pre)*av : (pre\TM -»¦ (pre о ^),OF-i@), X н
is (pre)*{E о Тм). The restriction
(pre)**v ¦ Q -> (pre ° Jr).CV-'@)
is an isomorphism of О м^-algebras.
D.18)
D.19)
D.20)
Proof, (a) It follows from (b).
(b) The kernel of (ргг)»а© is (pre)t(E о Тм) because of Theorem 4.9 (b).
The F-manifold in standard form has a global restricted Lagrange map L^r) c-*
T*M(r) -»¦ M(r) (the identification of its Lagrange fibration with T*Mir) -»•
M(r) is unique only up to shifts in the fibres).
The canonical projection F~l@) -*¦ L(r) is bijective (Corollary 4.5 (b)),
and then an isomorphism because F is holomorphic. It induces an isomor-
isomorphism (pre о ir),Of-i(o) = (тг<г))*О^(г). The composition with D.20) is the
isomorphism
a<r> : Q
from Lemma 3.17.
D.21)
D
Theorem 4.9 and Theorem 4.11 are translations to F-manifolds of state-
statements in [SK6, A.6)] [SK9, A.7)] for hypersurface singularities. In fact, K. Saito
essentially used D.20) to define the multiplication on Q for hypersurface
singularities.
The arguments in Lemma 4.12 and Theorem 4.13 are due to Lyashko [Lyl]
[Ly3] and Terao [Ter], see also Bruce [Bru].
Again(M, o, e, ?')isamassive F-manifold with Euler field of weight 1 and in
standard form. We choose a function t\:M~>C with e(t\) = 1 (e.g. t\ = — ? Л i,
cf. Lemma 3.20 (b)). This choice simplifies the formulation of the results in
Lemma 4.12. The vector fields in TMn will be identified with their (unique)

50
Discriminants and modality of F-manifolds
lifts in {X e Q | Х{Ц) = 0} CG С (рге)„Тм. The projection to TMcr, of all
possible lifts to M of vector fields in Mu' is
D.22)
Lemma 4.12 Let (M, о, е, Е, t\) be as above.
(a)
n-l
0
-t\-e
D.23)
(bj ?acft vector field in M(r) wft/cft /г/to to a vector field in (pre)*DerM (log V)
lifts to a unique vector field in D.23).
(c) The vector fields in M(r) which lift to vector fields in (pre)*DerM(logX>)
are tangent to the restricted bifurcation diagram B(r) = pre(B) С M(r) and
form the free 0Mm-module of rank n - 1
n-l
d(pr,)((r,e - ?H*) с Z)erMW(logB(r)). D.24)
k> 1
: (a) One sees inductively by multiplication with txe - E that for any
t\e -{he- E)°k 6 (prt),(E о Тм)
D.25)
holds. The inclusion he — E e ? and Liee(o) = 0 imply (t\e - ?)°* 6
therefore
л-l
*=o
tf e =
D.26)
Now the decomposition D.18) yields D.23).
(b) The map pre : V -+ M(r) is a branched covering of degree n (Definition
4.3), so
D.27)
*=o
is an isomorphism. Therefore any lift h ¦ e + X, h e (pre), OM, of X б Тмм can
be replaced by a unique lift h ¦ e + X with h 6 0^Zq Omw ff and (/г - А)Ь = 0.
If /г • e + X is tangent to D, then (ft - A)b = 0 is necessary and sufficient for
ft • e + X to be tangent to P.
Logarithmic vector fields
51
(с) A generic point p(r) 6 (B(r))reg has a preimage p e {Vsing)reg such that
the projection of germs pre : (Vsing, p) -* (B('\ p(r)) is an isomorphism
(Corollary 4.5 (a)). A vector field ft ¦ e + X, X e ТМю, which is tangent to
Vreg is also tangent to (T>Smg)reg- Then X is tangent to (B{r))reg. One obtains the
generators in D.24) by projection to TMm of the generators in D.23). ?
The set /CC) С /С С M is the set of points p 6 M such that (M, p) does not
decompose into 1- and 2-dimensional germs of F-manifolds (section 4.2).
Theorem 4.13 Let (M, o, e, E) be a massive F-manifold with Eulerfield E of
weight 1 and in standard form. Suppose that codim/CC) > 2.
Tlien the restricted bifurcation diagram B(r) is a free divisor and D.24) is an
equality.
Proof. In view of Lemma 4.12 (c) it is sufficient to show that any vector field
tangent to B(r) lifts to a vector field tangent to V.
The projection pre :V—B^>- M(r) —B(r) is a covering of degree л. For any
vector field X 6 TMw there exists a unique function hx 6 (рге)*От>-в such
that A • e + X is tangent to V - В if and only if ft Ь-в = hx.
One has to show that hx extends to a function in {pre\Ov if Xs
DerMM(logB(r)). Then the unique lift ft • e + X with h б ф^ OMv)t\ and
h\v = hx is tangent to V.
Let p be a point in the set
{p 6 B?^)^ | p(r) 6 (B(r))reg, pre : CD, p) -». (M<r\ p^) has degree 2}.
D.28)
Then the germ (D, p) is the product of (C"~2, 0) and the discriminant of the
germ of an F-manifold of type him) (m > 2) (Remark 4.2 (v)). One can find
coordinates (t\ ,t') = (tu..., tn) around p e M such that (D, p) С (М, p) ->•
(M(r), p(r)) corresponds to
({(fi, О I rf - f2m = 0}, 0) С (С", 0) -> (С1, 0), г н> г'. D.29)
Then (B(r), p{r)) = ({f' | Г2 = 0), 0). Obviously the vector fields tangent to
(B(r), p(r)) locally have lifts to vector fields tangent to (D, p). The function ft*
of a field X 6 DerMM(logB(r)) extends holomorphically to the set in D.28).
The complement in V oiVreg —V — B and of the set in D.28) has codimension
> 2 because of codim/CC) > 2. Therefore hx 6 {pre\Ov. D

52 Discriminants and modality ofF-manifolds
4.4 Isomorphisms and modality of germs of F-manifoIds
The following three results are applications of Theorem 3.19 for the Lyashko-
Looijengamap.Theywill be proved together. The tupIe((M, p), o, e, A)denotes
the germ of an F-manifold with the function germ Л : (M, p) ->¦ C" as addi-
additional structure. A map germ cp : (M, p) -> (M, p) respects Л if Л о ср = A.
Theorem 4.14 The automorphism group of a germ (M, p) of a massive F-
manifold is finite.
Theorem 4.15 Let(M,o,e, F)bea massive F-manifold with generating func-
function F and Lyashko-Looijenga map A : M ->• C". For any p\ e M the set
(qeM\ ((M, pi), o, e, A) S ((M, q), o, e. A)}
is discrete and closed in M.
Corollary 4.16 Let (M,o,e,E)bea massive F-manifold with Eulerfield E of
weight 1. For any p\ e M the set
{qeM\ ((M, p,), o, e, E) S ((M, q), о, е, Е)}
is discrete and closed in M.
Proof. Corollary 4.16 follows from Theorem 4.15. For Theorem 4.14, it suffices
to regard an irreducible germ of a massive F-manifold. The automorphisms of
an irreducible germ respect a given Lyashko-Looijenga map Л because of
Lemma 3.20 (a). So we may fix for Theorem 4.14 and Theorem 4.15 a massive
F-manifold (M, o, e) and a Lyashko-Looijenga map Л : M —> C.
The set
Д := {(P. p')eMxM\ A(p) = Л(р')}
has a reduced complex structure. It is a subset of(M-B)x(M-B)UBxS
and the intersection А П (M - B) x (M - B) is smooth of dimension n. This
follows from Theorem 3.19. ч
Now consider an isomorphism <p : ((M, p), o, e, A) -» ((M, p'), o, e, A).
The graph germ
(G(<p), (p, p')) := ({(?, cp(q)) e M x M | q near p}, (p, p'))
is a smooth analytic germ of dimension n and is contained in the germ
(Д, (p, p')). It meets А П (M - B) x (M - B). Because of the purity of
the dimension of an irreducible analytic germ, it is an irreducible component of
4.4 Isomorphisms and modality of germs of F-manifolds
53
the analytic germ (A, (p, p')). One can recover the map germ <p from the graph
gSrm(G(<p), (p, p'))- The germ (A, (p, p')) consists offinitely many irreducible
components. The case p = p' together with the remarks at the beginning of the
proof give Theorem 4.14.
For Theorem 4.15, we assume that there is an infinite sequence (p,-, ?>,-),¦ ещ
of different points p-, e M and map germs
<Pi : ((M, Pl), о, е, А) Д ((M, pi), o, e. A)
and one accumulation point рж e M. The set Д - В х В is analytic of pure
dimension n. It contains the germs (G(<Pi), (p\, Pi)) and the point (pi, p^).
We can choose a suitable open neighbourhood U of (p\, poo) in M x M and
a stratification
(J Sa = U П A-BxB
oft/ПА — В х В which consists of finitely many disjoint smooth connected
constructible sets Sa and satisfies the boundary condition: The boundary Sa - Sa
of a stratum Sa is a union of other strata.
The germ (G(<Pi), (p\, pi)) is an n-dimensional irreducible component of the
n-dimensional germ (A — В х В, (pi, pi)). There is a unique n-dimensional
stratum whose closure contains (G(<Pi), (pi, Pi))- If (Pi.Pi) € Sa then this
together with the boundary condition implies (Sa, (pi, pi)) С (<?fe)> (pi, Pi))-
The germ (G(<Pi), (pi, Pi)) is the graph of the isomorphism щ. Therefore it
intersects the germ ({p,} x M, (pb pi)) only in (pb pi); the same holds for
Now there exists at least one stratum Sao which contains infinitely many of
the points (pi, pi). The intersection of the analytic sets Sao and U П ({pi} x
M) contains these points as isolated points. This is impossible. The above
assumption was wrong. ?
In singularity theory there are the notions of/x-constant stratum and (proper)
modality of an isolated hypersurface singularity. One can define versions of
them for the germ (M, p) of an F-manifold (M, o, e) (massive or not massive):
The ix-constant stratum EД, p) is the analytic germ of points q e M such
that the eigenspace decompositions of TqM and TpM have the same partition
(cf. Proposition 2.5).
Theidempotent fields e\,..., et of the decomposition (M, p) = \\k=\(Mk, p)
into irreducible germs of F-manifolds commute and satisfy Lie,., (o) = 0 • o.
So the germs (M, q) of points q in one integral manifold of e\,..., ei are
isomorphic. This motivates the definition of the modality:
modM(M, p) := dimEM, p) -1. D.30)

54
Discriminants and modality of F-manifolds
Let (Sjf1, p) denote the /x-constant stratum of (Mk, p)\ Then Theorem 2.11
implies
mod (M, p) —
j) and
modM(Mb p).
D.31)
D.32)
For massive F-manifolds, Theorem 4.15 and Lemma 3.20 give more informa-
information:
Corollary 4.17 Let (M,p)= Y[[=i(Mk, P) be the germ of a massive F-
manifold and A : (M, p) —*¦ С a Lyashko-Looijenga map.
(a) There exist a representative 5M of the fi-constant stratum EД, р), а
neighbourhood U С С о/О and an isomorphism
U
D.33)
such that ifr~l([q] x U) is the integral manifold of eu ..., e/ which contains q.
Any subset of points in 5Д П Л (A(p)) with isomorphic germs of F-manifolds
is discrete and closed,
(b)
modM(M, p) = dimEM П A~\A(p)), p), D.34)
sup(modM(M, q) \ q near p) = dim(A~x(A(p)), p). D.35)
Proo/i (a) For / = 1, the existence of ф follows from the e-invariance of 5M
and from e{-\hx) = 1 (Lemma 3.20 (b)).
For arbitrary I, one uses D.31) and Lemma 3.20 (c): the maps A and
(A[1],..., Ли) have the same germs of fibres, especially
5Д П A-\A(p)) = Y\ S™ П AW"'(Aw(p)).
D.36)
A germ (M, q) has only a finite number of Lyashko-Looijenga maps with fixed
value at q (Lemma 3.20 (a)). The finiteness statement in Corollary 4.17 (a)
follows from this and Theorem 4.15.
(b) Equation D.34) follows from (a). A representative of the germ
(A~l(A(p)), p) is stratified into constructible subsets which consist of the
points q with the same partition for the eigenspace decomposition of TqM
(Proposition 2.5). A point q e A~'(A(p)) in an open stratum with maximal
dimension satisfies
modM(M, q) =
A~\A(p)), q)
= &m(A-\A{p)), q) = dim(A-1(A(p)), p). D.37)
4.4 Isomorphisms and modality of germs of F-manifolds 55
This shows
sup(modM(M, q)\qe A~\A(p)) near p) = dim(A-'(A(p)), p). D.38)
The upper semicontinuity of the fibre dimension of A gives D.35). ?
Remark 4.18 Gabrielov [Ga] proved in the case of isolated hypersurface sin-
singularities the upper semicontinuity of the modality,
modM(M, q) < modM(M, p) for q near p
D.39)
(and the equality with another version of modality which was defined by
Arnold). He used D.34), D.35), and a result of himself, Lazzeri, and Le, which,
translated to the F-manifold of a singularity (section 5.1), says:
EД П A-'(A(p)), p) = (Л-ЧЛ(р)), р).
D.40)
The inequality D.39) is an immediate consequence of D.34), D.35) and D.40).
But for other F-manifolds D.40) and D.39) are not clear.
In the case of the simple hypersurface singularities, the base of the semiuni-
versal unfolding is an F-manifold M = C" and the map A : M-B ->¦ <Cn-Vw
is a finite covering. Therefore the complement M — В is а К (л, 1) space and
the fundamental group is a subgroup of finite index of the braid group Br(n).
This is the application of Looijenga [Lol] and Lyashko [Arl] of the map A,
which led to the name Lyashko-Looijenga map.
It can be generalized to F-manifolds. We call a massive F-manifold M simple
if modM(M, p) = 0 for all p e M. This fits with the notions of simple hyper-
hypersurface singularities, simple boundary singularities, and simple Lagrange maps
([Gi2, 1.3, p. 3251]).
A distinguished class of simple F-manifolds are the F-manifolds of the finite
Coxeter groups (section 5.3 and [Lol][Arl][Lyl][Ly3][Gi2]). There are other
examples (Proposition 5.32 and Remark 5.33).
A Lyashko-Looijenga map of a massive F-manifold is locally a branched
covering if and only if M is simple (D.35) and Theorem 3.19). A detailed
proof of the following result had been given by Looijenga [Lol, Theorem 2.1]
(cf. also [Gi2, 1.4, Theorem 5]).
Theorem 4.19 Let (M, p) — (C, p) be the germ of a simple F-manifold with
fixed coordinates. Then, ife < sq for some sq, the space {z € C" | \z\ < e) — В
is а К (л, 1) space. Its fundamental group is a subgroup of finite index of the
braid group Br(n).

56 Discriminants and modality of F-manifolds
4.5 Analytic spectrum embedded differently
The analytic spectrum L С T*M of an F-manifold determines the multiplica-
multiplication on TM via the isomorphism (C.1) and B.2))
a : T
M
X н> a(x)\L.
D.41)
One can generalize this and replace L, T*M, and a by other spaces and other
1-forms. This allows F-manifolds to be found in natural geometric situations
and to be encoded in an economic way. Corollary 4.21 and Definition 4.23 are
the two most interesting special cases of Theorem 4.20.
Theorem 4.20 Let the following data be given:
manifolds Z and M, where M is connected and n-dimensional;
a surjective map nz '¦ Z -> M which is everywhere a submersion;
an everywhere n-dimensional reduced subvariety С С Z such that the restric-
restriction jtc '¦ С -> M is finite;
a I—form az on Z with the property:
any local lift X e Tz of the zero vector field 0 6 TM satisfies az(X)\c = 0.
D.42)
Then
(a) The map
&c : TM
X м- az(X)\c
D.43)
is welldefined; here X 6 Tz is any lift of X to a neighbourhood of С in Z.
(b) The image L С T*Mofthe map
q : С -> Т*М, г н* q(z) = (Хн ac(X)(z)) 6 T*ciz)M D.44)
is a (reduced) variety. The map q : С -> L is a finite map, the projections
ж : L —> M and tvq — ж о q are branched coverings. The composition of the
maps q : n*OL -> {пс\Ос and
a : TM
X н>- a\X)\L
D.45)
is Ac = q о a. All three are OM-module homomorphisms.
(c) The 1-forms a and az satisfy (q*a)|Cres = «zlc^- Therefore L is a
Lagrange variety if and only ifaz\cns is exact.
(d) The map a : 7д/ -> ntOi is an isomorphism if and only if
(i) the map ac is injective,
4.5 Analytic spectrum embedded differently
57
С (jtc)*Oc is multiplication invariant,
contains the unit \q 6 (лс)*Сс-
(ii) its image Лс
(Hi) the image Я
In this case Ac '¦ 1м -> (псХ^с induces a (commutative and associative and)
generically semisimple multiplication on Тм with global unit field and with
analytic spectrum L.
(e)Themap&c '¦ TM -> (лс)*0с provides Mwith the structure oj'a massive
F-manifold if and only ifotzlc^ " exact and the conditions (i)-(iii) in (d) are
satisfied.
Proof, (a) This follows from D.42).
(b) The equality dim С — n — dim M and nc finite imply that nc is open.
M is connected, thus же is a branched covering. Using local coordinates for
M and T*M one sees that q : С -> Т*М is an analytic map. The equality
jtc = я о q is clear and shows that q is finite. Then L = q(C) is a variety and it
is a branched covering. The equality ac = q о a follows from the definition of q.
(c) There is an open subset M@) С M with analytic complement M - M@)
such that л-cW') С С and n~\Mi0)) С L are smooth, nc : п^
Af@)and;r : л~\Мт) -> Af@) are coverings and q : Tr^W') ->
is a covering on each component of л~1(М^). Now &c = q о a implies
(d) The map q : я*О^ -> {жс)*Ос is an injective homomorphism of OM-
algebras. If a : TM -> ж%Оь is an isomorphism then (i)-(iii) are obviously
satisfied.
Suppose that (i)-(iii) are satisfied. Then a : Тм -> ж*О^ is injective with
multiplication invariant image a(TM) С ж+О^ and with \L 6 &(TM). The maps
a and ac induce the same (commutative and associative) multiplication with
global unit field on TM.
We have to show that this multiplication is generically semisimple with an-
analytic spectrum L. Then a : TM -> ж^Оь is an isomorphism and the proof of
(d) is complete.
If for each/? 6 M the linear forms in ж~1(р) С Т*М would generate a sub-
space of T*M of dimension < n then a would not be injective. So, for a generic
point p 6 M there exist n elements in ж~\р) С Т*М which form a basis of
T*M. We claim that я~'(р) contains no elements other than these: ж~1(р)
does not contain 0 6 T*M because of Ц 6 a(T^). From the multiplication
invariance of аGд/) one derives easily that ж~х(р) does not contain any further
elements.
This extends to a small neighbourhood U of the generic point p 6 M:
ж~1A]) consists of n sheets which form a basis of sections of T*M\ the map

58
Discriminants and modality of F-manifolds
л\ц :Tu -*¦ л*(п \U)) is an isomorphism and induces a semisimple multi-
multiplication on TM with analytic spectrum n~\U).
Then L is the analytic spectrum of the multiplication on TM because M is
connected.
(e) By (c) and (d) and Theorem 3.2. D
In Theorem 4.20 the map я : L -> M has degree n, but nc : С ->¦ A/ can
have degree > n; and even if пс : С ->• A/ has degree л the map q : С -*¦ L
does not need to be an isomorphism. Examples will be discussed below
(Examples 4.24, Lemma 5.17). But the most important special case is the
following.
Corollary 4.21 LetZ, M, nz, С С Z, az, ac, L, andqbeasin Theorem4.20.
Suppose that az\Creg is exact and &c '¦ TM -> {itc\Oc is an isomorphism.
Then q : С -*¦ L is an isomorphism and ac=qoa provides M with the
structure of a massive F-manifold with analytic spectrum L.
Proof. Theorem 4.20 (e) gives all of the corollary except for the isomorphism
q : С —*¦ L. This follows from the isomorphism q : n^Oi —у (ттс)*Ос and a
universal property of the analytic spectrum. ?
One can encode an irreducible germ of a massive F-manifold with data as in
Corollary 4.21 such that the dimension of Z is minimal.
Lemma 4.22 Let (A/, p) be an irreducible germ of a massive n-dimensional
F-manifold. LetmC TpM denote the maximal ideal in TpM.
(a) Then dim Z >n + dim m/m2 for any data as in Corollary 4.21.
(b) There exist data as in Corollary 4.21 for (A/, p) with dimZ = n +
dim m/m2 (the construction will be given in the proof).
Proof, (a) Ttcl(p) — 7tzl(p) П С consists of one fat point with structure
ring TPM. Its embedding dimension dim m/m2 is bounded by the dimension
dim 7rJ' (p) = dim Z — n of the smooth fibre п^\(р).
(b) One can choose coordinates (fb ..., tn) = (h, t') = t for (A/, p) with
e = ^ as usual and with
&-Ш=
m с TPM and
— | = т' с ТрМ
D.46)
D.47)
4.5 Analytic spectrum embedded differently 59
for m = 1+ dim m/m2. The dual coordinates on (T*M, T*M) are y{,..., yn,
the analytic spectrum is (cf. B.1))
L = [(y, t)\yx= 1, yiyj = Y^afjb'bk}- D.48)
Because of D.47) there exist functions bt e C{t'}[y2,..., ym] with
yi\L=bi(y2,...,ym,t')\L fori=m + l, ...,л. D.49)
We identify (A/, p) and (C\ 0) using (f,,..., *„) and define (Z, 0) = (С'" х
C", 0). The embedding
i: (Z, 0) = (Cm~l x C\ 0) ^ T*M, D.50)
(Л,..., *т_ь О н»- ()>, 0 = A, xi,..., xm-Ubm+l(x, t'),.... bn(x, t'), t)
provides canonical choices for the other data,
7TZ : (Z, 0) -»• (A/, p), (x, t) м- t, D.51)
С =1-41-), D.52)
1=2 /=m+l
The conditions in Corollary 4.21 are obviously satisfied.
D.53)
?
The notion of a generating family for a Lagrange map ([AGV1, ch. 19],
[Gi2, 1.4]) motivates us to single out another special case of Theorem 4.20.
Definition 4.23 Let Z,M,nz,C,az, and ac be as in Theorem 4.20 with
ссг\с„д exact and ac : TM -*¦ (яс)*Ос injective with multiplication invari-
invariant image ас(Тм) Э {lcb These data yield a massive F-manifold (A/, o, e).
A function F : Z —*¦ С is a generating family for this F-manifold if az =dF
and if С is the critical set of the map (F, nz): Z -*¦ С х A/.
There are two reasons for the name generating family:
A) The function F is considered as a family of functions on the fibres n^ip),
p e M.
B) The restriction of F to С is the lift of a generating function F : L -*¦ C,
i.e. F = F о q; so the 1-graph of F as a multivalued function on M is L.
In the case of a generating family the conditions D.42) and az exact are
obvious. The most difficult condition is the multiplication invariance of
It is not clear whether for any massive F-manifold M data (Z, nz, F) as
in Definition 4.23 exist. But even many nonisomorphic data often exist. We
illustrate this for the 2-dimensional germs him) of F-manifolds (section 4.1).

60
Discriminants and modality of F-manifolds
Examples 4.24 Always (Z, 0) = (C x C2, 0) and (M, p) = (C2, 0) with pro-
projection nz : (Z, 0) ->¦ (M, 0), (x, и, t2) н> (r,, t2) and e := S, := f, S2 :=
3
(a) С = {(x, t) | xm-2 - t\ = 0}, az = drx + xd?2.
These are data as in Corollary 4.21 for I2{m).
(b) Generating family F =t{+ f*(t2 - u2)kdu (k > 1), С = {(*, r) 112 -
x2 = 0},az|c=dF|c=(dri +c-x2*-1dr2)lcforsomec 6 С - {0}, ac(S2) ¦
These are data as in Definition 4.23 for I2Bk + 1), the map жс : С ->¦ Af has
degree 2, the map q : С ->¦ L is the normalization and the maximalization of
L (cf. [Fi, 2.26 and 2.29] for these notions).
(c) Generating family F = tx +xk+]t2 - Щх*+2 (к > 1), С = {(x, t) \ (t2 -
lc. These are data as in Definition 4.23 for I2Bk + 4), the map жс : С -+ M
has degree 2, the map q : С -+ L is the maximalization of L (for the missing
case /2D) compare Lemma 5.17).
(d) Generating family F = ц + f*(u2 - t2fudu (k > 1), С = {(x, t) | (t2 -
x2)x = 0}, az\Crtg = dF\Cl4 = (dr, + с ¦ xMdr2)|c^ for some с е С - {0},
ac(S2 - |cf* • SiJ = \c2tf ¦ lc. These are data as in Definition 4.23 for
h{2k + 2), the map nc : С -*¦ M has degree 3, the map q : С -*¦ L covers
one component with degree 1, the other with degree 2.
Chapter 5
Singularities and Coxeter groups
In this section several families of massive F-manifolds which come from sin-
singularity theory are studied. The most important ones are the base spaces of
semiuniversal unfoldings of hypersurface singularities. Three reasons for this
are: A) hypersurface singularities and their unfoldings are so universal objects;
B) their F-manifolds can be enriched to Frobenius manifolds (part 2); C) one
has a 1-1 correspondence between irreducible germs of massive F-manifolds
with smooth analytic spectrum and stable right equivalence classes of singular-
singularities (Theorem 5.6). This is covered in section 5.1. The discussion of boundary
singularities and their F-manifolds in section 5.2 is quite similar.
Sections 5.3 and 5.4 are devoted to finite irreducible Coxeter groups and their
F-manifolds and Frobenius manifolds. The discriminant in the complex orbit
space induces an F-manifold structure on the orbit space just as in Corollary 4.6.
This follows independently from work of Dubrovin and from results in singu-
singularity theory by Brieskorn, Arnold, O.P. Shcherbak, Givental. We extend work
of Givental in order to characterize these F-manifolds (Theorems 5.20, 5.21,
5.22) and use this to prove a conjecture of Dubrovin about the corresponding
Frobenius manifolds (Theorem 5.26).
In section 5.5 other families of F-manifolds with quite different properties
are constructed. A start is made on the classification of 3-dimensional germs of
massive F-manifolds.
5.1 Hypersurface singularities
A distinguished class of germs of massive F-manifolds is related to isolated
hypersurface singularities: the base space of a semiuniversal unfolding of
an isolated hypersurface singularity is an irreducible germ of a massive
F-manifold with smooth analytic spectrum (Theorem 5.3). In fact, there is a 1-1
61

62
Singularities and Coxeter groups
correspondence between such germs of F-manifolds and singularities up to
stable right equivalence (Theorem 5.6).
The structure of an F-manifold on the base space has excellent geometric
implications and interpretations (Theorem 5.4, Remarks 5.5). Many of these
have been known for a long time from different points of view. The concept
of an F-manifold unifies them. On the other hand, for much of the general
treatment of F-manifolds in this book the singularity case has been the model
case.
An isolated hypersurface singularity is a holomorphic function germ / :
(C"\ 0) -> (C, 0) with an isolated singularity at 0. Its Milnor number д 6 N is
the dimension of the Jacobi algebra Oe",o/(ff-, ¦ • •, ^-) — Oc.o/Jf-
The notion of an unfolding of an isolated hypersurface singularity goes
back to Thorn and Mather. An unfolding of / is a holomorphic function germ
F:(C"xC",0)^ (C, 0) such that F|c»,x(o) = /. The parameter space will
be written as (M, 0) = (<C\ 0).
The critical space (C, 0) С (Cm x M, 0) of the unfolding F - F(xu ... ,xm,
ti,...,tn)is the critical space of the map (F, pr): (Cm хМ,0)ч- (С х М, 0).
It is the zero set of the ideal
)
8хх'""дхя)
E.1)
with the complex structure Oc,o = Oc«xm,o/Jf\(C,0)-
The intersection С П (<С'И х {0}) = {0} is a point and (C, 0) is a complete
intersection of dimension n. Therefore the projection pr : (C, 0) -> (M, 0) is
finite and flat with degree /j, and Oc,o is a free CVo-module of rank pi.
A kind of Kodaira-Spencer map is the O^.o-linear map
Oc,o, X ь* X(F)|(C,0)
E.2)
where X is any lift of X e TM,o to (Cm x M, 0). Dividing out the submodules
шк,о • Тм,о and m^.o • Cc,o one obtains the reduced Kodaira-Spencer map
ac|o:7bAf-> Oc~,o/Jf. E.3)
All these objects are independent of the choice of coordinates. In fact, they even
behave well with respect to morphisms of unfoldings.
There are several possibilities to define morphisms of unfoldings (cf. Remark
5.2 (iv)). We need the following.
Let Ft : (Cm x M-,, 0) -> (С, 0), * = 1, 2, be two unfoldings of / with
projections prt : (<Cm x M,, 0) -»• (Af,-, 0), critical spaces C,, and Kodaira-
Spencer maps ac, ¦ A morphism from F\ to F2 is a pair (ф, фьа*е) of map germs
5.7 Hypersurface singularities
such that the following diagram commutes,
(<Cra хМ,,0)Л(С'" хМ2,0)
(Afb0) ^ (Af2,0),
and
63
E.4)
#lc«x{0} = id,
F\ = F2 о ф
E.5)
E.6)
hold. One says that F\ is inducedby (ф, фьа*е) from F2.
The definition of critical spaces is compatible with the morphism (ф, фь^е),
that is, фЧр2 = У/г, and (Сь 0) = ф~1((С2, 0)). Also the Kodaira-Spencer
maps behave well: the CV,,o-hriear maps
0Ml,o
Oci.o
are defined in the obvious way; their composition is
ac, = 0*I(C2,O) ° ac2 о йфЬсае.
E-7)
E.8)
E.9)
E.10)
Formula E.9) restricts to the identity on the Jacobi algebra of / because of
E.5). Therefore the reduced Kodaira-Spencer maps satisfy
c, lo = ac2lo
E.11)
An unfolding of / is versal if any unfolding is induced from it by a suitable
morphism. A versal unfolding F : (<Cm x M, 0) —> (C, 0) is semiuniversal if the
dimension of the parameter space (M, 0) is minimal. Semiuniversal unfoldings
of an isolated hypersurface singularity exist by the work of Thorn and Mather.
Detailed proofs can nowadays be found at many places, e.g. [Was][AGVl,
ch. 8].
Theorem 5.1 An unfolding F : (<Cm x M, 0) -> (C, 0) of an isolated hyper-
hypersurface singularity f : (<Cm, 0) —> (C, 0) is versal if and only if the reduced
Kodaira-Spencer map ado : TqM —> Oc»,o/Jf is surjective. It is semiuniver-
semiuniversal if and only г/ас lo и ап isomorphism.

64
Singularities and Coxeter groups
Remarks 5.2 (i) Because of the lemma of Nakayama ac lo is surjective (an
isomorphism) if and only if ac is surjective (an isomorphism).
(ii) A convenient choice of a semi universal unfolding F : (Cm xC,O) —>
(C, 0) is F{x\,..., xm, ti,...,tn) = f + ?f=i ntitj, where mu ..., wM €
Ос™,о represent a basis of the Jacobi algebra of /, preferably with
ТП\ = 1.
(iii) The critical space of an unfolding F : (Cm x C", 0) -> (C, 0) is reduced
and smooth if and only if the matrix (-X?-, т^г)@) has maximal rank m. This
OXj OXj OX'i Otfc
is satisfied for versal unfoldings.
(iv) In the literature (e.g. [Was]) one often finds a slightly different notion
of morphisms of unfoldings: An (r)-morphism between unfoldings Fi and Fi
as above is a triple (</>, <pbase, f) of map germs ф and фь^е with E-4) and E.5)
and r : (Af i, 0) -> (C, 0) with E.6) replaced by
= F2 о ф + т.
E.12)
The (r)-versal and (r)-semiuniversal unfoldings are defined analogously. They
exist because of the following fact [Was]: an unfolding F : (Cm x M, 0) ->
(C, 0) is (r)-versal ((T-)-semiuniversal) if and only if the map
C®T0M ->Oc»,o/Jf, (c, X) y+с
E.13)
is surjective (an isomorphism).
So one gains a bit: the base space of an (T-)-semiuniversal unfolding Fw has
dimension д — 1; if F(r) = F(r\x\,..., xm, t2,..., fM) is (T-)-semiuniversal
then f i + F^ is semiuniversal; between two semiunversal unfoldings t\ + F,(r)
and t\ + Fj of this form there exist isomorphisms which come from (r)-
isomorphismsof F,(r) and F2(r). (The relation between F(r) andti+F(r) motivates
the '(>")'» which stands for 'restricted'.)
On the other hand, one loses E.10). Anyway, one should keep (r)-semiuni-
(r)-semiuniversal unfoldings in mind. They are closely related to miniversal Lagrange maps
(see the proof of Theorem 5.6 and [AGV1, ch. 19]).
(v) One can generalize the notion of a morphism between unfoldings if one
weakens condition E.5): Let F, : (Cm x M,), 0) ->¦ (C, 0), i = 1, 2, be unfold-
unfoldings of two isolated hypersurface singularities Д and /2. A generalized mor-
morphism from Ft to F2 is a pair (ф, фь^е) of map germs with a commutative
diagram as in E.4) such that E.6) holds and 0|c»x{oj is a coordinate change
(between /1 and /2).
Then /1 and /2 are right equivalent. If the generalized morphism is invertible
then F\ and F2 are also called right equivalent.
Critical spaces and Kodaira-Spencer maps also behave well for generalized
morphisms; E.10) holds, in E.11) one has to take into account the isomorphism
5.7 Hypersurface singularities
of the Jacobi algebras of /1 and f2 which is induced by ф\с°х[0)-
65
The multiplication on the base space of a semiuniversal unfolding was first
defined by K. Saito [SK6, A.5)][SK9, A.3)].
Theorem 5.3 Let f : (Cm, 0) —> (C, 0) be an isolated hypersurface singularity
and F : (Cm xM,0)-> (C, 0) be a semiuniversal unfolding.
The Kodaira-Spencer map ас '¦ Тмл ~> C*c,o is an isomorphism and induces
a multiplication on Tm,q- Then (M, 0) is an irreducible germ of a massive F-
manifold with smooth analytic spectrum, and E := a^1 (F|c) is an Euler field
of weight 1.
Proof. The map ac : TMt0 -> 0c,o is an isomorphism because of Theorem 5.1
and Remark 5.2 (i). The critical space (C, 0) is reduced and smooth. One applies
Corollary 4.21 to (Z, 0) = (Cm x M, 0) and az = dF. The map q : (C, 0) ->
(L, 7t~l@)) is an isomorphism, and тг~'(О) is a point. Theorem 4.20 (c) shows
that F\c о q-1 is a holomorphic generating function. Therefore ? is an Euler
field of weight 1. ?
Theorem 5.4 Let f : (Cm,0)—> (C, 0) be an isolated hypersurface singularity
and Fi : (Cm x M,-, 0) -> (C, 0), i = 1,2, be two semiuniversal unfoldings.
There exists a unique isomorphism <p : (Mi, 0) —> (M2, 0) of F-manifolds
such that фьсие = <Pfor апУ isomorphism (ф, фьам) of the unfoldings F] and Fj.
Proof. The map фь^е '¦ W\, 0) -> (M2,0) is an isomorphism of F-manifolds
because of E.10).
Suppose that Ft = F2 and (Mi, 0) = (M2, 0). The tangent map of фъаж on
Г0М1 is dфbase\o = id because of E.11). The group of all automophisms of
(Mi, 0) as F-manifold is finite (Theorem 4.14). Therefore фЬа1е — id. ?
Remarks 5.5 (i) The rigidity of the base morphism фьа5е in Theorem 5.4 is in
sharp contrast to the general situation for deformations of geometric objects.
Usually only a part of the base space of a miniversal deformation is rigid with
respect to automorphisms of the deformation.
(ii) The reason for the rigidity is, via Theorem 4.14 and Theorem 3.19, the
existence of the canonical coordinates at generic parameters. The corresponding
result for singularities is that the critical values of F form coordinates on the
base at generic parameters. This has been proved by Looijenga [Lol].
(iii) Because of this rigidity the openness of versality (e.g. [Te2]) also takes
a special form: For any point t € M in a representative of the base space

66
Singularities and Coxeter groups
(M, 0) = (CM, O)ofasemhmiversal unfolding F, Theorem 2.11 yields a unique
decomposition (A/, t) = fliLi^b 0 into a product of irreducible germs of
F-manifolds. These germs (M^, t) are the base spaces of semiuniversal
unfoldings of the singularities of F\c»x{t]. The multigerm of F at Cm x {t} П С
itself is isomorphic - in a way which can easily be made precise - to a transversal
union of versal unfoldings of these singularities.
(iv) The tangent space T,M = 0*=1 T,Mk is canonically isomorphic to the
direct sum of the Jacobi algebras of singularities of F |e» * (/) • The vector in T, M
of the Euler field E is mapped to the direct sum of the classes of the function
Лс»х{/} in these Jacobi algebras. A result of Scherk [Sche2] says:
The Jacobi algebra Oc,o/Jf of an isolated hypersurface singularity
f : (Cm, 0) -> (C, 0) together with the class [/] e Oc-,o/J/ determines
f up to right equivalence.
This result shows that the base space M as an F-manifold with Euler field E de-
determines for each parameter t e M the singularities of F|c»x{r) up to right equiv-
equivalence and also the critical values. Theorem 5.6 will give an even stronger result,
(v) The eigenvalues of Eo : T,M -> T,M are the critical values of F|c»x(r).
Therefore the discriminant of the Euler field E is
V = {t e M | (det(?o))(/) = 0} = яс(С Л F~\0))
E.14)
and it coincides with the classical discriminant of the unfolding F.
All the results of section 4.1 apply to this discriminant. Of course, many of
them are classic in the singularity case.
For example, Theorem 4.1 and the isomorphism q : С -> L from Corollary
4.21 yield an isomorphism between the development T> С РТ*М of the dis-
discriminant and the smooth variety С П F~'@) which has been established by
Teissier [Te2]. Implicitly it is also in [AGV1, ch. 19].
The elementary way in Corollary 4.6 in which the discriminant and the unit
field determine the Jacobi algebras seems to be new. But the consequence from
this and Scherk's result that the discriminant and the unit field determine the
singularity (up to right equivalence) is known (compare below Theorem 5.6
and Remark 5.7 (iv)).
Arnold studied the relation between singularities andLagrange maps [AGV1,
ch. 19]. His results (cf. also [Hoe], [Phi, 4.7.4.1, pp. 299-301], [Ph2], [Wir,
Corollary 10]) together with those of section 3.4 yield the following correspon-
correspondence between unfoldings and certain germs of F-manifolds.
Theorem 5.6 (a) Each irreducible germ of a massive F-manifold with smooth
5.1 Hypersurface singularities
67
analytic spectrum is the base space of a semiuniversal unfolding of an isolated
hypersurface singularity.
(b) Suppose, Ft : (C'"( x Af,-, 0) -> (C, 0), i = 1, 2, are semiuniversal un-
unfoldings of singularities f : (C"\ 0) -> (C, 0) and (p:(Mu0)-> (M2, 0) is
an isomorphism of the base spaces as F-manifolds. Suppose thatm\ < m2.
Then a coordinate change f : (C', 0) -+ (C', 0) exists such that
/,(*!,..., xmi) + x2mi+l + ..-+x2m2 = f2{xu ..., xm2) о ф E.15)
and an isomorphism (ф, фьа*е) of the unfoldings Fy + x^i+1 -I h хгтг and
Fivty exists with
and
E-16)
Proof, (a) The restricted Lagrange map of the germ of a massive F-manifold
with smooth analytic spectrum is a miniversal germ of a Lagrange map with
smooth Lagrange variety (section 3.4). Arnold [AGV1, 19.3] constructed a
generating family F(r) = Fir\x, t2,..., t^) for it. Looking at the notions of
stable maps and generating families in [AGV1, ch. 19], one sees: F(r) is an
(r)-semiuniversal unfolding of F(r){x, 0) (cf. Remark 5.2 (iv)). The unfolding
tx 4- f M is a semiuniversal unfolding of F{r)(x, 0). Its base space is the given
germ of a massive F-manifold.
(b) The unfolding F,- is isomorphic to an unfolding Ц + F/r)(*i,. ..,xmi,
t2,.. ¦, tft.) as in Remark 5.2 (iv) over the same base. Then F;<r) is an (r)-
semiuniversal unfolding and a generating family for the restricted Lagrange
map of the F-manifold (Af,-, 0).
The isomorphism (p : (Af i, 0) -> (Af2, 0) induces an isomorphism of the re-
restricted Lagrange maps. Then the main result in [AGV1, 19.4] establishes a
notion of equivalence for F\r) and F^, stable 7?+-equivalence, which yields
the desired equivalence in Theorem 5.6 (b) for F| and F2. ?
Remarks5.7 (i) Two isolated hypersurface singularities/; : (C"",0) -> (C, 0)
with mi < m 2 are stably right equivalent if a coordinate change \jr : (СШ2,0) ->
(С2, 0) with E.15) exists. Furthermore they are right equivalent if m, = m2.
The splitting lemma says:
An isolated hypersurface singularity f : (Cm, 0) -*¦ (C, 0)withr :=m-
rank(g^?-)@) is stably right equivalent to a singularity g : (C, 0) ->
(C, 0) with rank(g|^)@) = 0; this singularity g is unique up to right
equivalence.

68
Singularities and Coxeter groups
(For the existence of g see e.g. [SI, D.2) Satz], the uniqueness of g up to right
equivalence follows from Theorem 5.6 or from Scherk's result (Remark 5.5
(iv)).)
(ii) Theorem 5.6 gives a 1-1 correspondence between isolated hypersurface
singularities up to stable right equivalence and irreducible germs of massive
F-manifolds with smooth analytic spectrum.
But the liftability of an isomorphism cp : (M\, 0) -» (Л/2, 0) to unfoldings
which is formulated in Theorem 5.6 (b) is stronger. The 1-1 correspondence it-
itself already follows from Theorem 5.6 (a) and Scherk's result (Remark 5.5 (iv)).
(Hi) The proof of Theorem 5.6 (a) is not very difficult. If (M, 0) is an irre-
irreducible germ of a massive F-manifold with analytic spectrum (L, А.) С Т*М,
then a sufficiently generic extension of a generating function on (L, A.) to a
function on (T*M, A.) is already a semiuniversal unfolding over (M, 0). A ver-
version different from [AGV1,19.3] of the precise construction is given by Pham
[Phi, 4.7.4.1, pp. 291-301], following Hormander [Hoe].
(iv) Theorem 5.6 (b) follows also from [Ph2] (again following Hormander)
and from [Wir, Corollary 10]. To apply Wirthmiiller's arguments one has to
start with the discriminant V and the unit field. Pham [Phi] [Ph2] starts with the
characteristic variety. That is the cone in T*M of the development 2? с ?Т*М
of the discriminant.
A semiuniversal unfolding F : (Cm x M, 0) -» (C, 0) yields data as in
Corollary 4.21 for the germ (Л/, 0) of an F-manifold:
(Z, 0) = (Cm x M, 0), az = dF.
E.17)
The semiuniversal unfolding F is also a generating family of (M, 0) as a germ
of an F-manifold in the sense of Definition 4.23.
The following observation says that these two special cases Corollary 4.21
and Definition 4.23 of the general construction of F-manifolds in Theorem 4.20
meet only in the case of unfoldings of isolated hypersurface singularities.
Lemma 5.8 LetZ, M,nz,C,oiz,ac,andF : Z -> С satisfy all the properties
in Corollary 4.21 and Definition 4.23.
Then С is smooth. For any point p e M the multigerm F : (Z,C П
1 '
1 z (?)) -*¦ ^ is isomorphic to a transversal product ofversal unfoldings of the
singularities ofF\n^\p) (cf. Remark5.5 (Hi)). The irreducible germs (Mb p)
of F-manifolds in the decomposition (M, p) = Ц1к=](Мк, р) are base spaces
of semiuniversal unfoldings of the singularities o
^ '
(p).
5.2 Boundary singularities
69
Proof. The isomorphism ac : TM -*¦ (,nc)*Oc of Corollary 4.21 restricts at
p e M to a componentwise isomorphism of algebras
(Jacobi algebra of F\(nz' (p), z)).
One applies Theorem 5.1.
?
5.2 Boundary singularities
The last section showed that germs of F-manifolds with smooth analytic spec-
spectrum correspond to isolated hypersurface singularities. The simplest nonsmooth
germ of an analytic spectrum of dimension n is isomorphic to
We will see that irreducible germs of massive F-manifolds with such an analytic
spectrum correspond to boundary singularities (Theorem 5.14). Boundary sin-
singularities had been introduced by Arnold [Ar2]. Because of the similarities to
hypersurface singularities we will take things forward exactly as in section 5.1.
We always consider a germ (Cm+1,0) with coordinates xo,...,xm together
with the hyperplane H := {x 6 Cm+1 | x0 = 0} of the first coordinate. A
boundary singularity (/, H) is a holomorphic function germ / : (Cm+1, 0) ->¦
(C, 0) such that / and f\H have isolated singularities at 0. It can be considered
as an extension of the hypersurface singularities / and f\H-
Its Jacobi algebra is
Eл8)
and its Milnor number д =
[Sz, §2])
Я) := dimOc^.o/J/.H satisfies ([Ar2, §3],
E.19)
An unfolding of (/, Я) is simply a holomorphic function germ F : (Cm+1 x
C, 0) -> (C, 0) such that F|Cm+1 x {0} = /, that is, an unfolding of/. Again
we write the parameter space as (M, 0) = (C\ 0).
But the critical space (C, 0) С (Cm+1 x M, 0) of F as unfolding of the
boundary singularity (/, Я) is the zero set of the ideal
dF 3F dF\
J E20)

70
Singularities and Coxeter groups
with the complex structure Oc,o = 0o»+' xM,o/jf,h\(c,O) (cf. [Sz]). Forgetting
the complex structure, (C, 0) is the union of the critical sets (C(l), 0) of F and
(CB), 0) of F\hxm as unfoldings of hypersurface singularities.
For the same reasons as in the hypersurface case the projection pr : (C, 0) —>
(M, 0) is finite and flat with degree /u. and Oc 0 is a free CV^-module of rank
д. The 1-form
az := -
E.21)
on (Z, 0) := (C"+I x M, 0) gives rise to a kind of Kodaira-Spencer map
ac : TMi0 -»¦ Oc.o, X м- az(#)l(c,o)- E.22)
where X is any lift of X e TM,oto(Z, 0).Itinduces a reduced Kodaira-Spencer
map
ac|o : T0M -»• Oe»xfi/Jf,H. E.23)
The ideal Jf, h and the maps ac and ac |o behave well with respect to morphisms
of unfoldings, as we will see.
A morphism between two unfoldings F\ and F2 as in section 5.1 of a bound-
boundary singularity (/, H) is a pair (ф, фь^е) of a map germ with E.4)-E.6) and
additionally
E.24)
ф(Н x МО С Я x M2.
Then the first entry of ф takes the form xq • unit 6 Oz,o- Using this one can
see with a bit more work than in the hypersurface case that the critical spaces
behave well with respect to morphisms:
ф*]Рг = JF, and (C,,0) = Ф~\(С2, 0)). E.25)
Also the Kodaira-Spencer maps behave as well as in the hypersurface case.
The OWb0-linear maps AфЬа8е, ac2, and ф*\(с2,0) are defined as in E.7)-E.9);
again one finds
and
ас, = Ф*\(сг,о) о ас2 о йфЬа]
ac,lo =
E.26)
E.27)
Versal and semiuniversal unfoldings of boundary singularities are defined anal-
analogously to the hypersurface case and they exist.
5.2 Boundary singularities
71
Theorem 5.9 [Ar2] An unfolding F : (Cm+1 x M, 0) -> (C, 0) of a boundary
singularity (/, H), f : (C+1, 0) -> (C, 0), is versal if and only if the reduced
Kodaira-Spencer map ac|o •' T0M ->• Ос™+1,о/^/,я is surjective. It is semi-
universal if and only if ado is an isomorphism.
Remarks 5.10 (i) The map ado is surjective (an isomorphism) if and only if
ac is surjective (an isomorphism).
(ii) The function F(x0,..., xm, t{,..., fM) = / + ??., m,-f,- is a semiuni-
semiuniversal unfolding of the boundary singularity (/, H) if m \,..., mM 6 Oo"+I 0
represent a basis of Oc"+',o/^/,«-
(iii) The critical space of an unfolding F : (Z,0) = (Cn+1 xM,0)-+ (C, 0)
of a boundary singularity (/, H) is reduced and isomorphic to ([(x, у) е
C2 I xy = 0}, 0) x (С", 0) if and only if f?,..., |?- represent a generating
system of the vector space mz^AOto) + m| 0). This is equivalent to the non-
degeneracy condition
rank
гапк
bxabxj
d2F
Ьх;Ьх;
32F
Эх0Э1к
32F
@) = m + 1
E.28)
(cf. [DD]). It is satisfied for versal unfoldings.
(iv) As in Remark 5.2 (v) for hypersurface singularities, one can define
generalized morphisms between unfoldings of right equivalent boundary sin-
singularities. Again the critical spaces and Kodaira-Spencer maps behave well.
Theorem 5.11 Let F : (Cm+1 x M, 0) -> (C, 0) be a semiuniversal unfolding
of a boundary singularity (/, #).
The Kodaira-Spencer map &c • ^м,о ~~*" Oc,o is an isomorphism and induces
a multiplication on 7м,о- Then (M, 0) is an irreducible germ of a massive F-
manifold with analytic spectrum isomorphic to ({(x, y) 6 C2 | xy = 0}, 0) x
(О*,0). The field E := ac(F|c) is an Euler field of weight 1.
Proof. Similar to the proof of Theorem 5.3. One wants to apply Corollary
4.21 and has to show that <xz\creg is exact. The critical space (C, 0) as a set
is the union of the smooth zero sets (CA\ 0) of Jf and (CB), 0) of the ideal
. The definition E.21) of az shows
(*o, f^, ¦...,
azl(c«),0) = dF|(C(o,0) for i = 1, 2.
E.29)
(
Therefore az\creg is exact and F\c о q~' is a holomorphic generating function
of the analytic spectrum. ?

72
Singularities and Coxeter groups
Theorem 5.12 Let Ft : (C+1 x M,, 0) -> (C, 0), i = 1,2, be two semiuni-
versal unfoldings of a boundary singularity (/, H).
There exists a unique isomorphism <p : (M\, 0) —> (M2, 0) of F-manifolds
such that fybase — <Pforany isomorphism (ф, фьа*е) of the unfoldings F\ and F2.
Proof: Similar to the proof of Theorem 5.4.
D
Remarks 5.13 (i) Let F : (C+1 x M, 0) -* (C, 0) be a semiuniversal unfold-
unfolding of a boundary singularity (/, H) with critical space (C, 0) = (CA\ 0) U
(CB), 0). For any t e M the points in Cm+1 x [t} П (C, 0) split into three classes:
The hypersurface singularities of F\Cm+1 x [t) in CA) - CB), the hypersurface
singularities of F\H x {t) in CB) — CA), and the boundary singularities of
F|Cm+1 x {t} in СA)ЛС<2).
The algebra Oc|Cm+l x {0} is the direct sum of their Jacobi algebras. The
reduced Kodaira-Spencer map at t e M is an isomorphism from T, M to this
algebra.
Hence the multigerms of F at Cm+1 x [t] П CA) and of F\H x M at H x
{t) П (CB) — CA)) together form a transversal union of versal unfoldings of
these hypersurface and boundary singularities.
The components {Mk, t) of the decomposition (M, t) = \^к=\Шк, t) into
irreducible germs of F-manifolds are bases of semiuniversal unfoldings of the
hypersurface and boundary singularities.
(ii) The eigenvalues of Eo : T,M —y T,M are by definition of E the values
of F on Cm+1 x {t} П С The discriminant of the Euler field is
V = [t e M | (det(?o))(f) = 0} = 7TC(C П F~' @)). E.30)
This is the union of the discriminants of F and F\H x Mas unfoldings of
hypersurface singularities and it coincides with the classical discriminant of F
as an unfolding of a boundary singularity [Ar2][Sz]. All the results of section
4.1 apply to this discriminant.
Nguyen huu Due and Nguyen tien Dai studied the relation between boundary
singularities and Lagrange maps [DD]. Their results together with section 3.4
yield the following correspondence between unfoldings of boundary singular-
singularities and certain germs of F-manifolds.
Theorem 5.14 Let(M, 0) be an irreducible germ of a massive F-manifold with
analytic spectrum (L, X) isomorphic to
({U, y) € C2 | xy = 0}, 0) x (С, 0)
and ordered components (LA), X) U (LB), k) = (L, X).
5.2 Boundary singularities
73
(a) There exists a semiuniversal unfolding F of a boundary singularity such
that the base space is isomorphic to (M, 0) as F-manifold and the isomorphism
q : (C, 0) -*¦ (L, A.) maps C(l) to L(i\
(b) Suppose, Fi : (C"+1 x M,-,0) -»¦ (C, 0), i = 1,2, are semiuniversal
unfoldings of boundary singularities (/|, Я,) and <p : (Mt, 0) -> (M2, 0) is аи
isomorphism of the base spaces as F-manifolds. Suppose thatm\ < m2. '
Then a coordinate change i/r : (СШ2+1, 0) -> (Cm2+1, 0) with i/((H2,0)) =
(Hz, 0) exisfs suc/z that
fx(x0,.. .,xm) +xl1+l + • • • +xl2 = /2(*
isomorphism (ф, фъа5е) of the unfoldings F}
of boundary singularities exists with
0,... ,xm2) о
E.31)
-I h jc2
* =
and
E.32)
Froq/: (a) In [DD, Proposition 1] an unfolding F : (Cm+1 x M, 0) -> (C, 0)
with nondegeneracy condition E.28) of a boundary singularity is constructed
such that F is a generating family for LA) С T*M and F \ H x M is a generating
family for LB).
One can show that there are canonical maps C(l) —v L(l) which combine to
an isomorphism q : С —у L with &c = q о a (as in Theorem 4.20). Then
the Kodaira-Spencer map ac : Тм,о -у Ос,о is an isomorphism and F is
a semiuniversal unfolding of a boundary singularity. (Implicitly this is also
contained in [DD, Theoreme]). Because of ac = q о a its base is (M, 0) as
F-manifold.
(b) [DD, Proposition 3]. D
Remarks 5.15 (i) Two boundary singularities fx : (C""+1,0) -> (C, 0) with
mi < m2 are stably right equivalent if a coordinate change i/r as in Theorem
5.14 (b) exists. Furthermore they are right equivalent if m\ = m2. A splitting
lemma for boundary singularities is formulated below in Lemma 5.16.
(ii) Theorem 5.14 gives a 1-1 correspondence between boundary smgularities
up to stable right equivalence and irreducible germs of massive F-manifolds with
analytic spectrum (L, X) = {{(x, y) € C2 \xy - 0}, 0) x (C2,0) and ordered
components (LA), k) U (LB), X) = (L, X).
(iii) Interchanging the two components of (L, X) corresponds to a duality
for boundary singularities which goes much further and has been studied by I.
Shcherbak, A. Szpirglas [Sz][ShSl][ShS2], and others.

74 Singularities and Coxeter groups
Lemma 5.16 (Splitting lemma for boundary singularities)
A boundary singularity (/, H)with f : (C'"+1, 0) -+ (C, 0)andH = {x\xo =0}
is stably right equivalent to a boundary singularity g : (C+1, 0) —*¦ (C, 0) in
@) E.33)
г + 1 = maxl 2; т + 1 - rank
coordinates. The boundary singularity g is unique up to right equivalence.
Proof. Existence of g: The group G = Z2 acts on (C!+1, 0) by (x0, xt,...,
xm) \-± (±x0, X], ...,xm). Boundary singularities on the quotient (Cm+1, 0)
correspond to G-invariant singularities on the double cover, branched along H
([AGV1, 17.4]).
One applies an equivariant splitting lemma of Slodowy [SI, D.2) Satz] to the
G-invariant singularity f(Xg, xt xm). The nondegenerate quadratic part of
the G-invariant singularity in splitted form does not contain x\ because / is
not smooth.
Uniqueness of g: This follows with Theorem 5.14 (b). D
The following two observations give some information on generating families
in the sense of Definition 4.23 for the F-manifolds of boundary singularities.
The first one gives a distinguished generating family and is essentially well
known. The second one explains why B2 = /2D) is missing in Example 4.24 (b).
Lemma 5.17 (a) Let F : (Z, 0) = (Cm+1 xM,0)-> (C, 0) be a semiuniversal
unfolding of a boundary singularity (/, H).
Then the function F : (Z, 0) = (Cm+I x M, 0) ->¦ (C, 0) with F(x, t) =
F(x%, x\,...,xm,t) is ageneratingfamily for the germ (M, 0) of an F-manifold.
The finite map q : С -у L = LA) U LB) from its critical set С to the analytic
spectrum L has degree 2 on LA) and degree 1 on LB). The branched covering
С ^ Mhas degree 2fi(f) + n(f\H)-
(b) Let (M, 0) be a germ of a massive F-manifold with analytic spectrum
(L, X) ~ ([(x, y) € C2 | xy = 0}, 0) x (С, 0). There does not exist a gen-
generating family F : (Z, 0) —у (С, 0) with critical set С such that the canonical
map q : С —у L is a homeomorphism.
Proof: (a) Consider the branched covering лгс : Z -> Z, (xo,..., xm, t) н>
(Xq, x\ ,..., xm, t) which is induced by the action (x0, ...,xm,t) н* (±*о.
X\,..., xm, f) of the group G = Z2 on Z. The composition F = F о лс is an
unfolding of the G-invariant singularity F|(Cm+I x{0}, 0), infact, semiuniversal
5.5 Coxeter groups and F-manifolds
ithin the G-invariant unfoldings (cf. [SI, D.5)]). The ideals
/ /3F \ 3F 3F \
\ \dxQ ) dx{ dxm )
75
E.34)
nd tvqJf.h have the same zero sets С = UqX{C).
Comparison of a^ — dFonZandaz = — |j-dxo+dFonZ(formulaE.21))
shows that the map a^ : Тм,o —*¦ Cc.o factorizes into the Kodaira-Spencer map
shows th p ^ , .
ас '- 1~м,о -> Oc,o and the map Jr?|(c,o) : Oc,o -* C»c,o-
lilii irit
с - 1м,о > Oc,o and the map Jr?|(c,o) c,o c,o
Therefore ag is injective with multiplication invariant image and induces the
correct multiplication on Тц,й- The rest is clear.
(b) Assume that such a generating family F exists. The analytic spectrum
(L, X) is its own maximalization. Therefore the homeomorphism q is an iso-
isomorphism.
Then ac = q ° a (cf. Definition 4.23 and Theorem 4.20) is an isomorphism.
We are simultaneously in the special cases Definition 4.23 and Corollary 4.21
of Theorem 4.20. By Lemma 5.8 (L, X) is smooth, a contradiction. D
5.3 Coxeter groups and F-manifolds
The complex orbit space of a finite irreducible Coxeter group is equipped with
the discriminant, the image of the reflection hyperplanes, and with a certain dis-
distinguished vector field (see below), which is unique up to a scalar. Together they
induce as in Corollary 4.6 the structure of an F-manifold on the complex orbit
space (Theorem 5.18). This follows independently from [Du2][Du3, Lecture
4] and from [Gi2, Theorem 14].
In fact, both give stronger results. Dubrovin established the structure of a
Frobenius manifold. This will be discussed in section 5.4. Givental proved that
these F-manifolds are distinguished by certain geometric conditions (Theorem
5.21). With one additional argument we will show that the germs of these
F-manifolds and their products are the only germs of simple F-manifolds whose
tangent spaces are Frobenius algebras (Theorem 5.20). This complements in
a nice way the relation between Coxeter groups and simple hypersurface and
boundary singularities.
We will also present simple explicit formulas for these F-manifolds which
are new for Я3 and H\ (Theorem 5.22).
A finite Coxeter group is a finite group W of linear transformations of the
Euclidean space W generated by reflections in hyperplanes. Each Coxeter group
is the direct sum of irreducible Coxeter groups. Their classification and descrip-
description can be found in [Co] or [Bou]. They are An (n > 1), Dn (n > 4), Ef,, ?7,

76
Singularities and Coxeter groups
Et, Bn (n > 2), F4, G2, H3, Щ, I2(m) (m > 3) with A2 = /2C), B2 = /2D),
H2 := /2E), G2 = /2F).
The Coxeter group W acts on C" = R" ®R С and on C[X],..., xn], where
x\,... ,х„ are the coordinates on C". The ring С [xi,..., xn]w of invariant poly-
polynomials is generated by n algebraically independent homogeneous polynomials
P\,..., Pn. Their degrees d; := deg P,- are unique (up to ordering). The quo-
quotient C/ W is isomorphic to C" as an affine algebraic variety. The C*-action
and the vector field ?,. xt j^ on the original С induce a C*-action and a vector
field ^,ditjj^ on the orbit space C/ W = C. The image in the orbit space
of the union of the reflection hyperplanes is the discriminant V of the Coxeter
group.
Suppose for a moment that W is irreducible. Then there is precisely one
highest degree, which is called the Coxeter number h. The degrees can be
ordered to satisfy
di = h > d2 > ... > dn-i >dn=2,
di+dn+1-i =h + 2.
The vector field e := ^- is unique up to a scalar.
E.35)
E.36)
Theorem 5.18 The complex orbit space M := C/W = С of a finite irre-
irreducible Coxeter group W carries a unique structure of a massive F-manifold
with the unit field e — ^- and the discriminant T>. The discriminant V corre-
corresponds to the Eulerfield
E.37)
of weight 1.
Proof. The uniqueness follows from Corollary 4.6.
The existence follows from Dubrovin's result ([Du2][Du3, Lecture 4], cf.
Theorem 5.23) or Givental's result [Gi2, Theorem 14] together with Theorem
3.16.
Below in Theorem 5.22 we will follow Givental and reduce it to classical re-
results on the appearance of discriminants in singularity theory ([Bril] [Ar2] [Ly2]
[ShO]). D
Remarks 5.19 (i) Corollary 4.6 gives probably the most elementary way in
which e and V determine the multiplication on the complex orbit space M =
C/ W = C, at least at a generic point: the e-orbit of a generic point p e M
5.3 Coxeter groups and F-manifolds
11
intersects V transversally in n points. One shifts the tangent spaces of V at these
points with the flow of e to TpM. Then there exists a basis eit... ,е„ of TPM
such that Ya=\ e< = e and such that the hyperplanes 0,-^ С ¦ e,-, ; = 1,..., n,
are the shifted tangent spaces of V. The multiplication on TpM is given by
ei oej = Sijej.
(ii) The unit field e = ^ is only unique up to a scalar. The flow of the Euler
field respects the discriminant V and maps the unit field e and the multiplication
to multiples, because of Lie?(e) = —e and Lie?(o) = o.
Therefore the isomorphism class of the F-manifold (M, о, е, Е) is indepen-
independent of the choice of the scalar.
(iii) The complex orbit space of a reducible Coxeter group W is isomorphic
to the product of the complex orbit spaces of the irreducible subgroups. The
discriminant decomposes as in Remark 4.2 (v). Now any sum of unit fields for
the components yields a unit field for C/W. The choices are parameterized by
(С*I'гг ™*«га"'. But the resulting F-manifold is unique up to isomorphism. It is
the product of F-manifolds for the irreducible subgroups. This F-manifold and
its germ at 0 will be denoted by the same combination of letters as the Coxeter
group.
Theorem 5.20 Let ((M, p), o, e) be a germ of a massive F-manifold.
The germ ((M, p), o, e) is simple and TpM is a Frobenius algebra if and
only if((M, p), o, e) is isomorphic to the germ at 0 of an F-manifold of a finite
Coxeter group.
This builds on the following result, which is a reformulation with section 3.4
of a theorem of Givental [Gi2, Theorem 14]. Theorems 5.20, 5.21, and 5.22
will be proved below in the opposite order. Some arguments on Я3 and Щ in
the proof of Theorem 5.21 will only be outlined.
Theorem 5.21 (Givental) (a) The F-manifold of a finite irreducible Coxeter
group is simple. The analytic spectrum (L, X) of its germ at 0 is isomorphic to
({(x,y) eC2|jc2=: /},0) x (Си-\0) withr= lforAn, Dn, En, r = 2for
Bn, F*. r = 3/or Я3, H4 and r = m - 2 for him).
(b) An irreducible germ of a simple F-manifold with analytic spectrum iso-
isomorphic to a product of germs of plane curves is isomorphic to the germ at 0
of an F-manifold of a finite irreducible Coxeter group.
Finally, we want to present the F-manifolds of the finite irreducible Coxeter
groups explicitly with data as in Corollary 4.21. We will use the notations of
Corollary 4.21. The following is a consequence of results in [Bril][Ar2][Ly2]

78
Singularities and Coxeter groups
[ShO] on the appearance of the discriminants of Coxeter groups in singularity
theory.
Theorem 5.22 (a) The germs at 0 of the F-manifolds of the Coxeter groups
An, D,,, E,,, Bn, F4 are isomorphic to the base spaces of the semiuniversal
unfoldings of the corresponding simple hypersurface singularities A,,, Dn, En
and simple boundary singularities Bn (or Cn) and F4.
(b) For the F-manifolds (M, о, е) = (С, 0, e) of the finite irreducible Cox-
Coxeter groups, a space Z with projection itz : Z -*¦ M, a subspace С С Z and
a 1-form az will be given such that the map
X н* az(X)\c
E.38)
is welldefined and an isomorphism of Qм-algebras. The space С is isomor-
isomorphic to the analytic spectrum of (M, o, e). The Euler field is always E =
\ E"=i diUw,- The discriminant V С М is V = лс(ас(Е)~1 (О)).
(i) А„, В„.'H3, him) :
Z = CxM = CxC" with coordinates (x, t) = (x, t\,..., tn),
az=dti l
, t) :=
- \)x
.n-1.
h },
Bn:C = {(x,t)eZ\x- (xn~x - g) = 0},
H3: С = {(x, f)eZU3-? = 0},
him) :C = {(x,t)eZ\x2- 72m~2 = 0}.
(it) D4, F4, H4 :
Z = C2 x M = C2 x C4 with coordinates (x, y, t) = (x, y, t\,..
az — dt\ + xdt2 + ydt-i + xydt4,
h(x, y, t) := h + ytt, b(x, У, t) ¦¦= h + xt4,
D4:C = {(x,y,t)e Z\x2 + f2=0,y2 + h = 0},
F4:C = {(x,y,t)eZ\x2 + r2= 0, у2 + Г32 = 0},
H4:C = {{x, y, t)eZ\x2 + r2= 0, y2 + t? = 0}.
(iii)Dn,E6,E7,E8:
Z — C2 x M = C2 x С with coordinates (x, y, t) ;= (x, y, t\,..
F : Z -*¦ Си semiuniversal unfolding of F\C2 x {0},
Dn: F=xn-'
E6: F =
., tn),
= |f =0},
yh +x2t4 + xyts + x2yt6,
5.3 Coxeter groups and F-manifolds
79
F = хъу
F = x5 +
у3 + t\
/ +1\
+ xt2 + yh + x2t4 + xyt5 + x3f6 + x4t7,
xt2 + yt3 + x2t4 + xyts + x3t6 + x2yt7
Proof of Theorem 5.22: (a) One can choose a semiuniversal unfolding F =
f(xi,..., xm) + Yl"=] m'fi of the hypersurface or boundary singularity which
is weighted homogeneous with positive degrees in all variables and parameters.
There is an isomorphism from its base space C" to the complex orbit space of
the corresponding Coxeter group which respects the discriminant, the Euler
field, and the unit field ([Bril][Ar2]). It also respects the F-manifold structure
(Corollary 4.6).
(b) Part (i) for /2(m) is Remark 4.24 (a). Part (i) for А„, В„ and part (ii) for
D4 follow with (a), with semiuniversal unfolding as in (a) for the singularities
__i_x»+i (A,,), (_ij.» + yit н = {x = 0}) {В„), \хг + |v3 (D4). Also (iii)
follows with (a).
The same procedure gives for the boundary singularity F4 with equation
(|y2 + |x3, H = {y = 0}) the data in (ii) with critical set
2 + T2 = 0, у2
С = {{x,y,t)€Z\x2
уГ3 = 0}.
E.39)
It is a nontrivial, but solvable, exercise to find compatible automorphisms of Z
and M which map С to С andaz to «z modulo /c^z. Independently of explicit
calculations, the proof of Theorem 5.21 will show that the data (Z,ctz, C) in
(ii) correspond to F4.
The data in (i) for Щ and in (ii) for H4 can be obtained from results of O.P.
Shcherbak [ShO, pp. 162 and 163] (cf. also [Gi2, Proposition 12]) (for Я3 one
could use instead [Ly2]). The unfoldings
= f
Jo
xJdu+tl+xt2+x2t3
of ZN and
p«4 = / (и2
Jo
t3 +xt4fdu
t\ + xt2
E.40)
E.41)
of Es have only critical points with even Milnor number and are maximal with
this property. Their discriminants are isomorphic to the discriminants of the
Coxeter groups H3 and H4. The unfoldings are generating families in the sense
of Definition 4.23 for the F-manifolds of the Coxeter groups H3 and H4. We
will determine the data in (ii) for H4 from F#4; the case of H3 is similar.
Consider the map
ф : C2 x C4 -> Z = C2 x С4, (х,у, t) ь» (х,у, t),
E.42)

80
Singularities and Coxeter groups
y(x, y, t) := / 2(«2 + f3 + xt4)du = -y3 + 2(f3 + хц)у, E.43)
and observe
ytA)dx + (y2 + f3
+ At i + xdt2 + y&h + xydt4,
9
2
3 = (У2
Therefore
V
ф*(а2) = dF - ^dbc - ^-dy
dx dy
and the image under ф of the reduced critical set С г of FHi is
E.44)
E.45)
E.46)
= \(x, y, t) € Z|3*2 + t2 + yt4 = 0, —y2 + h +xt4 = 0 .
16
E.47)
An automorphism
s~x
Z-> Z, (x, y, t\,t2, h, f4) (-»• {r~lx, s~xy, tu rti, st3, rst4) E.48)
for suitable r, s € C* maps ф(Ср) to С and respects 7tz and az, together with
the induced automorphism M -+ M. ?
Sketch of the proof of Theorem 5.21: (a) Consider the data in Theorem 5.22
(b). The Euler field on M = С is E = { ?(. rf,f, ^. The coefficients of the
Lyashko-Looijenga map Л : M —*¦ C" are up to a sign the symmetric polyno-
polynomials in the eigenvalues of Ea. Because of Lie^Eo) = Eo, the coefficient Л,-
is weighted homogeneous of degree i with respect to the weights (f-, • ¦ •. ^)
for (ti,..., tn). The Lyashko-Looijenga map is a branched covering of degree
E.49)
and (M, o, e) is simple (Corollary 4.17 (b)). The analytic spectrum is isomorphic
toC.
(b) The dimension dim(A/t, p) of an irreducible germ in the decomposition
(Л/, р) = П*=|(^*> Р) °f a germ °f a massive F-manifold is equal to the
intersection multiplicity of T* M with the corresponding germ (L, A.*) of the
analytic spectrum L. This number will be called the intersection multiplicity
IM(Xk).
5.3 Coxeter groups and F-manifolds
81
(SM(g), q) denotes for any q e M the /x-constant stratum through g (section
4.4), and /(<?) the number of irreducible components of (M, q). For any sub-
variety ScLwe have the estimates
max(l(q)\q € яE)) < и + 1 - min(IM(a)\cr e 5), E.50)
max(dim(SM(<7), ?) | ? 6 5) > dim 5, E.51)
max(modM(M, q) \ q € S) + n + 1 - min(/M(<r) | ст € S) > dim5. E.52)
Therefore, if M is simple then
min(/M(<r) | a 6 S) < n + 1 - dim S E.53)
for any subvariety 5 С L.
Now suppose that ((Л/, р), о, e) is an irreducible germ ofa simple F-manifold
and that ф : (M, p) ~+ П?=](^1> ^) is an 18отофЫ8т to a product of germs
of plane curves (they are necessarily plane because of Proposition 3.7).
If at least two curve germs were not smooth, e.g. (Cn_i, 0) and (С„, 0), then
the intersection multiplicities IM(p) for points p in S\ :— Ф'ЧПы?^, О) х
{0}) would be at least 4; but dim(Si, p) = n — 2, a contradiction to E.53). So,
atmost one curve, e.g. (Cn, 0), is not smooth.
The irreducible germs of F-manifolds which correspond to generic points of
7гEг) for S2 := 0~1(П!'=11(^1'0) x {0}) are at most 2-dimensional because of
E.53). Therefore
€<C2 \x2 = yr},0)
E.54)
for some r e N. If r > 4 and n > 3 then the set of possible intersection multi-
multiplicities for points in S2 has a gap at 3 and a subvariety S3 С S2 exists with dim
S3 = n - 2 and min(/A/(cr) | a € S3) > 4 [Gi2, p. 3266], a contradiction to
E.53). Therefore r € {1,2, 3} or л < 2.
If г e {1,2} then (A/, p) is the base space of a semiuniversal unfolding of
a hypersurface singularity (r = 1, Theorem 5.7, [AGV1, ch. 19]) or boundary
singularity (r = 2, Theorem 5.15, [DD]). Simplicity of their F-manifolds corre-
corresponds to simplicity of the singularities. The simple hypersurface singularities
are An, Dn, Еь, Ei, E$ [AGV1]. The simple boundary singularities are Bn, Cn,
and F4 [Ar2] [AGV1]. The boundary singularities Bn and Cn are dual boundary
singularities and have isomorphic discriminants and F-manifolds.
The details of the case r = 3 [Gi2, pp. 3269-3271] are difficult and will
not be given here. In that case the set of possible intersection multiplicities for
points in S2 has a gap at 5. If n > 6 then a subvariety S4 С S2 exists with
dimS4 = n - 4 and min(IM(a) \ a € S4) > 6, a contradiction to E.53). The

82
Singularities and Coxeter groups
case r = 3 and n = 3 corresponds to #3, the case r = 3 and n — 4 corresponds
to #4. D
Proof of Theorem 5.20: It is sufficient to consider an irreducible germ {{M, p),
0, e). If it corresponds to a Coxeter group then it is simple (Theorem 5.21 (a))
and TpM is a Frobenius algebra (Theorem 5.22 (b)).
Suppose that (M, p) is simple and that TpM is a Frobenius algebra. We
will show by induction on the dimension n — dim M that the analytic spec-
spectrum (L, A) is isomorphic to {{(x, y) e C2 \ x2 = /'}, 0) x (C"~', 0) for some
reN.
This is clear for n = 2. Suppose that n > 3. The maximal ideal of TpM
is called m. The socle Апп^аДш) of the Gorenstein ring TpM has dimension
1, therefore Апптрм0^) 5 m and m2 ф 0. In the equations for the analytic
spectrum (L, А) С 7^M one can eliminate fibre coordinates which correspond
to m2 С ТрМ: the embedding dimension of (L, A) is
m
embdim(L, A) < л + dim —- < In — 2
E.55)
(Lemma4.22). Then (L, A) = (C2,0) x (L", A") (Proposition 3.7). There exists
A2 6 L close to A such that (L, A2) = (L, A) and^(A2)isnotinthee-orbitof p.
Now for all q near p, but outside of the e-orbit of p, the germ (M, «jO is
reducible because of modM(M, p) = 0. For all q near p the germ (M, <y) is
simple and TqM is a Frobenius algebra (Lemma 2.2).
One can apply the induction hypothesis to the irreducible component of
(M, n{X.2)) which corresponds to A2. Its analytic spectrum (Z/, A') is isomorphic
to a product of a smooth germ and a curve as above. Now (L, A) = (L, A2) ==
(C"-dimZ-', 0) x (I/, A'). One applies Theorem 5.21 (b). D
5.4 Coxeter groups and Frobenius manifolds
K. Saito [SK3] introduced a flat metric on the complex orbit space of a finite
irreducible Coxeter group. Dubrovin [Du2][Du3, Lecture 4] showed that this
metric and the multiplication and the Euler field from Theorem 5.18 together
yield the structure of a massive Frobenius manifold on the complex orbit space
(Theorem 5.23).
The Euler field has positive degrees. Dubrovin [Du2][Du3, p. 268] conjec-
conjectured that these Frobenius manifolds and products of them are the only massive
Frobenius manifolds with an Euler field with positive degrees.
We will prove this conjecture (Theorem 5.25). Theorem 5.20, which builds
on Givental's result (Theorem 5.21, [Gi2, Theorem 14]), will be crucial.
5.4 Coxeter groups and Frobenius manifolds
83
We use the same notations as in section 5.3. A metric on a complex manifold is
a nondegenerate complex bilinear form on the tangent bundle. The flat standard
metric on C" is invariant with respect to the Coxeter group W and induces a fiat
metric g on M — V. Dubrovin proved the following with differential geometric
tools [Du2][Du3, Lecture 4 and pp. 191 and 195].
Theorem 5.23 (Dubrovin) Let W be a finite irreducible Coxeter group with
complex orbit space M = C/ W, Euler field E, a unit field e, and a multipli-
multiplication о on M as in Theorem 5.18.
The metric g on M —V with
g(X, Y) := g(E о X, Y)
E.56)
for any (local) vector fields X and Y extends to aflat metric on M and coincides
with K. Saito'sflat metric. (M, о, е, Е, g) is a Frobenius manifold. The Euler
field satisfies
UeE(g) = (l + !) g. E.57)
There exists a basis of flat coordinates z\,. ¦ ¦, Zn on M with Zi@) = 0 and
E.58)
Remarks 5.24 (i) K. Saito (and also Dubrovin) introduced the flat metric g in
a way different from formula E.56): The metrics g and g on M — V induce
two isomorphisms T(M - V) -> T*(M - V). The metrics g and g are lifted
with the respective isomorphisms to metrics g* and g* on the cotangent bundle
T*(M - V). Then
g* = 1ле,(Г)
E.59)
([Du3, pp. 191 and 195]). (Here g* and g* are considered as @, 2)-tensors.) K.
Saito introduced g with the formula E.59).
(ii) Closely related to E.56) and E.59) is ([Du3, pp. 191 and 270])
J2IT1 • IT* = 8*(dQudQ2) = ifi(dQ, оdQ2). E.60)
frf dXj dxt
Here Qi, Q2 6 C[jci, ..., xn]w are W-invariant polynomials; dgi and dQ2
are inteфreted as sections in T*M; the multiplication о is lifted to T*M with
the isomorphism TM —> T*M induced by g; is is the contraction of a 1-form
withE.

84
Singularities and Coxeter groups
The first equality is trivial. Equation E.60) is related to Arnold's convolution
of invariants ([Ar3][Gil]).
(iii) A Frobenius manifold as in Theorem 5.23 for a finite irreducible Coxeter
group is not unique because the unit field and the multiplication are not unique.
Contrary to the F-manifold, it is not even unique up to isomorphism. There is
one complex parameter between (M, o, e) and (M, g) tobechosen: (M, о, e, Е,
с -g) respectively (M, c-o, c~[ e, E, g) is a Frobenius manifold for any с е C*.
(iv) We consider only Frobenius manifolds with an Euler field which is
normalized by Lie? (o) = 1 ¦ о (compare Remark 2.17 (c)). The product ]~[ M,- of
Frobenius manifolds (M,-, o,-, e,-, ?,, g,) also carries the structure of a Frobenius
manifold if Lie^,. (g,-) = Dg, holds with the same number DeC for all i. This
follows from Proposition 2.10, Theorem 2.15 and Remark 2.17 (c) (compare
also[Du3,p. 136]).
(v) Especially, the complex orbit space C/W of a reducible Coxeter group
can be provided with the structure of a Frobenius manifold if the irreducible
Coxeter subgroups have the same Coxeter number.
The Frobenius manifold is not unique. The different choices are parame-
parameterized by (С*)!'"™*»™"»! in the obvious way (cf. the Remarks 5.19 (iii) and
5.24 (iii)).
The spectrum of a Frobenius manifold (M, o,e, E, g) is defined as follows
(cf. Remark 9.2 e)). The Levi-Civita connection of the metric g is denoted by
V. The operator V? :TM -*¦ Тм, Хн Vx?, acts on the space of flat fields
([Du3, p. 132], [Manl, p. 24]) and coincides there with -ad E. The set of its
eigenvalues {w\,..., wn) is the spectrum ([Manl]). If —ad E acts semisimple
on the space of flat fields then there exists locally a basis of flat coordinates
Z\ zn with
E =
9
iZi +n)—
E.61)
for some r, e C.
The following was conjectured by Dubrovin ([Du2][Du3, p. 268]).
Theorem 5.25 Let ((M, p), о, е, Е, g) be the germ of a Frobenius manifold
with the following properties:
generically semisimple multiplication;
Lie?(o) =l-o and Lie?(g) = D ¦ g;
* = ^z,-
for a basis of flat coordinates Zi with Zi(p) = 0;
positive spectrum (w\,... ,wn), that is, to,- > Ofor all i.
E.62)
5.4 Coxeter groups and Frobenius manifolds 85
Then (M, p) decomposes uniquely into a product of germs at 0 of Frobenius
manifolds for certain irreducible Coxeter groups. The Coxeter groups have all
the same Coxeter number h = -pZ\-
Proof. As in the proof of Theorem 5.21 (a), the hypotheses on the Euler field
show that the Lyashko-Looijenga map Л : (Мь р) -»¦ С is finite and that
the F-manifold (M, o, e) is simple. One applies Theorem 5.20 and Theorem
5.26.
D
Theorem 5.26 Let ((M, p), о, е, Е, g) be the germ of a Frobenius manifold
such that ((M, p), o, e, E) is isomorphic to the germ at 0 of the F-manifold of
a finite Coxeter group with the standard Euler field.
Then the irreducible Coxeter subgroups have the same Coxeter number and
((M, p), o, e, E, g) is isomorphic to a product of germs at 0 of Frobenius man-
manifolds for these Coxeter groups.
Proof: First we fix notations. W is a finite Coxeter group which acts on V = C"
and respects the standard bilinear form. The decomposition of W into I irre-
irreducible Coxeter groups W\,..., Wi corresponds to an orthogonal decomposi-
decomposition V = 0?=1 Vt. The choice of n algebraically independent homogeneous
polynomials Pi,...,Pn € C[jci ,..., xn]w identifies the quotient M = V/ W
with C". The quotient map xj/ : V -*¦ M decomposes into a product of quotient
maps yjfk : Vk -»• Vk/ Wk = Mk. The F-manifold M = Y\'k=i Mk is the product
of the F-manifolds Mk.
Setting e := ?"_, xt -^ on У and ek := e| vt, the standard Euler field Ek on
Mk is Ek — ^-Airk{ek). Here hk is the Coxeter number of Wk. The Euler field
on M is
^=Е?'=Ею'^- E-63)
{wi,..., wn} is the union of the invariant degrees of Wk, divided by hk.
Now suppose that g is a flat metric on the germ (M, 0) such that ((M, 0), o, e,
E, g)isagerm of aFrobenius manifold withLie?(g) = Dg. Consider a system
of flat coordinates zi,..., Zn of (M, 0) with z,@) = 0. The space 0"=1 С ¦ ^
of flat fields is invariant with respect to adE ([Du3, p. 132], [Manl, p. 24])
and the space of affine linear functions С • 1 ф 0"=1 С • z; С Ом,о is invariant
with respect to E. Because E vanishes at 0 even the subspace 0"=1 С • z,- is
invariant with respect to E.
The weights wi,..., wn of E are positive. Therefore the coordinates zi, • • •.
zn can be chosen to be weighted homogeneous polynomials in C[f i,..., tn] of

86
Singularities and Coxeter groups
degree wu ...,wn. Thus the spectrum of the Frobenius manifold is {ш|
w,,}. It is symmetric with respect to ~, because of Lie?(g) = D ¦ g; hence
1 + y. = ? for all fc = 1,...,/. The Coxeter numbers are all equal, h :=h\ =
•••=/!;, the Euler field E is
E = j^dfie). E.64)
It remains to show that g is induced as in Theorem 5.23 from a metric on V which
it the orthogonal sum of multiples of the standard metrics on the subspaces V*.
The operatorU = Eo : TM -» TM is invertible onM-P. The metric g on
M-V with
#(Х,У):=*(&Г1Ш,1Г) E.65)
is flat ([Du3, pp. 191 and 194], [Manl]). It lifts to aflat metric g on V-f~\V).
We claim that g extends to a flat metric on the union f~{(V) of the reflection
hyperplanes.
It is sufficient to consider a generic point p in one reflection hyperplane. Then
the ^-orbit of V(p) intersects V in n points; there exist canonical coordinates
Mi,..., un in a neighbourhood of \//(p) with e,- о ej = S^ei, g(e,, ej) = 0 for
л
E = usei + ^(m,- + r,)e,- for some r,- G C*.
i=2
(Р,^р)) = ({и|щ=0},0).
The map germ -ф : (У, р) —> (M, i/(p)) is a twofold covering, branched along
(V, f(p)), and is given by (Й1,..., un) н> (гГ,2, и2,...,ип) = (мь ..., и„)
for some suitable local coordinates m~i ,..., un on (V, p). Then
!1 зй[)~8
о e,, e,) =
,, e,),
~/ Э
1
M,+r,ev
~/ Э 9 \
т^г.^г =0 for»
for/>2,
So g extends to a nondegenerate (and then flat) metric on V.
The Coxeter group W acts as a group of isometries with respect to g. It
remains to show that the vector space structure on V which is induced by
~g (and 0 € V) coincides with the original vector space structure. Then g is
an orthogonal sum of multiples of the standard metrics on the subspaces V*,
5.5 3-dimensional and other F-manifolds
87
because each W-invariant quadratic form is a sum of Wk -invariant quadratic
forms on the subspaces V* and they are unique up to scalars.
Let?be the vector field on V which corresponds to the C*-action of the vector
space structure induced by g. Then Li&gg = 2 ¦ g. Because of 1лъЕ(Ы) = U,
LieE(g) = (l + f)g,and? = jdi/f(s) we also have Lieeg = 2 ¦ g for the vector
field ?, which corresponds to the C*-action of the old vector space structure.
The differences—? satisfies Lies-j(g) = 0 and is a generator of a 1-parameter
group of isometries. As it is also tangent to the union of reflection hyperplanes,
it vanishes. The vector field e = "e determines a unique space of linear functions
on V and a unique vector space structure. D
5.5 3-dimensional and other F-manifolds
The F-manifolds in sections 5.1-5.3 were special in several aspects: the analytic
spectrum was weighted homogeneous and a complete intersection. Therefore
an Euler field of weight 1 always existed, and the tangent spaces were Frobenius
algebras. Furthermore, the stratum of points with irreducible germs of dimen-
dimension > 3 had codimension 2.
Here we want to present examples with different properties. A partial classi-
classification of 3-dimensional germs of massive F-manifolds will show that already
in dimension 3 most germs are not simple and do not even have an Euler field
of weight 1. Examples of germs (M, p) of simple F-manifolds such that TpM
is not a Frobenius algebra will complement Theorem 5.20.
First, a construction which is behind the formulas for An, Bn, #3, /2(m) in
Theorem 5.22 (b)(i) provides many other examples.
Proposition 5.27 Fix the following data: (M, 0) = (C, 0),
(Z, 0) = (C, 0) x (M, 0) with coordinates (x, t) = (x, h,..., tn\
the projection nz : (Z, 0) -> (M, 0),
the 1-form az := dti + xdt2 H h xn~ldtn on Z, .; ;,
the function 72(x, t) := t2 + 2xt3 H \-(n - l)xn~\,
an isolated plane curve singularity (or a smooth germ) f : (C2, 0)
with f(x, 0) = xn ¦ unit g C{x},
the subvariety С := {(x, t) e Z \ f(x, T2) = 0} С Z.
(a) The map
(C, 0)
ac :
¦ Oc,o,
az(X)\c
E.66)
(X is a lift of X to Z) is welldefined and an isomorphism of О'м ^-modules.
The germ (M, 0) with the induced multiplication on Tuss is an irreducible
germ of a massive F-manifold. Its analytic spectrum is isomorphic to

88
Singularities and Coxeter groups
(С, 0) = (С"-1, 0) х (/"'(О), 0). For each t e M the tangent space T,M is is-
omorphic to a product of algebras C{x}/(xk) and is a Frobenius algebra.
(b)An Euler field of weight 1 exists on (M, p) if and only if the curve singu-
singularity f(x, y) is weighted homogeneous.
(c) Suppose that mult/ = n. Then the caustic is К = {t e M | t2 = 0}. The
germ (M, t) is irreducible for all t 6 1С, so the caustic is equal to the ^.-constant
stratum of(M, 0). The modality is mod^M, 0) = n -1 (the maximal possible).
Proof, (a) The 1-form ctz is exact on Creg because of daz = dxd72. One can
apply Corollary 4.21.
(b) Corollary 3.8 (b).
(c) For t2 = 0 fixed we have f(x, T2) = x" ¦ unit e C{x, t3,..., tn]. Thus the
projection nc : С -*¦ M is a branched covering of degree n, with ж^' ({t \ t2 =
0}) = {0} x {t | t2 = 0} and unbranched outside of {t \ t2 = 0). The analytic
spectrum is isomorphic to C. ?
Remarks 5.28 (i) The function T2 is part of a coordinate system on T*M for a
different Lagrange fibration: the coordinates
У\ - У\, Уг - Уг, ft - У! ~ y'2~l for i > 3, E.67)
• 1- (и — l)v?~2fn, Ti = t; for J > 3
satisfy
yidti = da.
E.68)
The analytic spectrum of an F-manifold as in Proposition 5.27 is
L = {(y, t)eT'M\yi = l, f(y2, Г2) = 0,у!=0 for i > 3}. E.69)
It is a product of Lagrange curves.
(ii) Another different Lagrange fibration is behind the formulas for D4, F4, #4
in Theorem 5.22 (b)(ii). There are many possibilities to generalize the construc-
construction of the above examples.
In dimension 3, there exist up to isomorphism only two irreducible commu-
commutative and associative algebras,
й(|) := С{х)/(хг) and E.70)
Qa):=C{x,y}/(x2,xy,y2y, E.71)
and 6(l) is a Frobenius algebra, gB) not.
5.5 3-dimensional and other F-manifolds
89
Theorem 5.29 Let (M, p) be an irreducible germ of a 3-dimensional massive
F-manifold with analytic spectrum (L,X) С Т*M.
(a) Suppose TpM = g(I). Then (L, X) has embedding dimension 3 or 4 and
(L, X) = (C2, 0)x(C", 0) for a plane curve (C, 0) С (С2, 0)vwY/imult(C", 0) <
3. An Eulerfield of weight 1 exists if and only if(C, 0) is weighted homogeneous.
(b)Suppose TpM = Qmand(L, X) = (C2, 0)x(C, 0)wf/zmult(C, 0) < 3.
Then ((M, p), o, e) is one of the germs A3, Вт,, Щ.
(с)Suppose TpM = Q{v>and(L,X) = (C2, 0)x(C", 0)vwf/imult(C", 0) = 3.
Then the caustic 1С is a smooth surface and coincides with the ц,-соп-
stant stratum; that means, TqM = 2A) for each q 6 1С. The modality is
mod^(M, p) = 1 (the maximal possible).
(d) Suppose TpM = QB). Then (L, X) has embedding dimension 5 and
(L, X) = (C, 0) x (L(r), 0). Here (L(r), 0) is a Lagrange surface with embedding
dimension 4. Its ring Oz,w,o is a Cohen—Macaulay ring, but not a Gorenstein
ring.
Proof, (a) One chooses coordinates (fb t2,
Lemma 4.22) with e = -щ and
9f2
C- — =
1 for (M, p) (as in the proof of
m С TpM, E.72)
m2 с ТРМ. E.73)
The dual coordinates on (T*M, T*M) are y\,...,yn. There exist functions
e0, au a2, b0, b\, b2 e C{t2, f3} with
\yi =
= b2y\
y2 =a2y\
E.74)
The Hamilton fields of the smooth functions y\ — \ and уъ - Yll=o Ъ\У2 are щ
and|-+.. .in T*M. They are tangent to L. Therefore (L, X) = (C2, 0)x(C, 0)
with (C", 0) = (L, X) П T*M (cf. Proposition 3.7). The statement on the Euler
field is contained in Corollary 3.8 (b).
(b) We have mult(C", 0) < 2; and the intersection multiplicity of (C, 0) with a
suitable smooth curve is 3. So, (C, 0) is either smooth or a double point or a cusp.
In the first two cases, one can apply the correspondence between F-manifolds
and hypersurface or boundary singularities (Theorem 5.6 and Theorem 5.14)
and the fact that A3, Biy and C3 are the only hypersurface or boundary singu-
singularities with Milnor number 3.

90
Singularities and Coxeter groups
Suppose (С, 0) is a cusp and ((M, p), o, e) is not #3. Then it is not simple
becauseofTheorem5.20.Theц.-constantstratumS^ = {q e M \ TqM = Qm)
is more than the e-orbit of p. It can only be the image in M of the surface of
cusp points of L, because at other points q close to p the germ (M, q) is A\ or
A\ x A2.
So at each cusp point X' of L the intersection multiplicity of T*(y)M and
(L, X') is 3. This property is not preserved by small changes of the Lagrange
fibration (e.g. as in Remark 5.28 (i)). But Givental proved that a versal Lagrange
map is stable with respect to small changes of the Lagrange fibration [Gi2,
p. 3251, Theorem 3 and its proof]. This together with section 3.4 yields a
contradiction.
(c) In this case, at each point A/ (close to X) of the surface of singular points
of L the intersection multiplicity of T*^M and (L, X') is 3 and the map germ
я : (L, A.') —*¦ (M, n(X')) is a branched covering of degree 3. This implies all
the statements.
(d) If the embedding dimension of (L, X) is < 4 then (L, X) = (C2, 0) x
(C, 0) for some plane curve (C, 0) by Proposition 3.7. Then(L, A.) is a complete
intersection and the tangent spaces TPM are Frobenius algebras.
So if TPM = <2B) then the embedding dimension of (L, X) is 5. The ring
Ow),o is a Cohen-Macaulay ring because the projection L(r) —*¦ M(r) of the re-
restricted Lagrange map is finite and flat. It is not a Gorenstein ring because TpM
is not a Gorenstein ring. ?
The next result provides a complete classification and normal forms for those
irreducible germs (M, p) of 3-dimensional massive F-manifolds which satisfy
TpM = gA) and whose analytic spectrum consists of 3 components. Part (a)
gives an explicit construction of all those F-manifolds.
Theorem 5.30 (a) Choose two discrete parameters p2, рз e Nwithp2 > рз >
2 and choose рз — l holomorphic parameters (go, g\,..., gPi-2) G С* х C3~2
with go ф 1 if pi = рз-
Define (M, 0) := (C3, 0),
(Z, 0) := (C, 0) x (M, 0) with coordinates (x,tu t2, t3) = (x, t),
fx:=Ch:=t^,2f3:=k-S,
С ¦¦= ULi c<- := UL K*. 0 e Z I x = f|} с Z,
b2(t2, t3) := (ръ-g+h- f^) • (рз • g + h ¦ |^ -p2 ¦ t^2~n)-1
(b2 is a unit in C{t2, t3\),
b[(t2, ?з) .'= —b2 ¦ p2 • ^2 >
the 1-form az '¦= dfi + xdt2 + (b2x2 + b\x)dt3 on Z.
5.5 3-dimensional and other F-manifolds
91
(i) Then
- for i = 1,2,3, E.75)
«zlc, = d(f, + f,)\c, for i = 1, 2, 3. E.76)
&c '¦ 1~м,о —> Octo, X (->¦ &z(X)\c E.77)
Chj The map
(X is a lift ofX to Z) is welldefined and an isomorphism ofOMfi-modules. The
germ (M, 0) with the induced multiplication on 7м,о is an irreducible germ of
a massive F-manifold with TqM =. <2A). Its analytic spectrum is isomorphic to
(C,0) = (C2,0) x(C',0), with
(C, 0) = (C, 0) П ({(*, t)\ti=t3= 0}, 0).
E.78)
(Hi) The caustic is K. = {t e M \ t2 = 0} and coincides with the bifurcation
diagram В and with the ^-constant stratum.
(iv) The functions t\ + fi\cr i = 1> 2, 3, combine to a function F : С —> С
which is continuous on С and holomorphic on Creg = С П {(x, t) \t2 ф 0}. The
Eulerfield E on M — fC with лс\м-к.(Е) = Лм-к is
-t2^)~. E.79)
dtj dt
The following conditions are equivalent:
(a) The function F is holomorphic on С and E is holomorphic on M,
(P) one has рз = 2 or (p2 = рз > 3 and gt = Ofor 1 < i < рз — 2),
(у) the curve (C, 0) is weighted homogeneous.
(b) Each irreducible germ (M, p) of a massive F-manifold such that TPM =
2A> and such that(L, X) has 3 components is isomorphic to a finite number of
normal forms as in (a). The numbers p2 and рз are determined by (L, X). The
number of isomorphic normal forms is < 2p2 ifp2 > рз and < 6p2 ifp2 = рз.
Proof, (a) (i) Direct calculation.
(ii) The map ас is an isomorphism because b2 is a unit in €,{t2, ?з}. One can
apply Corollary 4.21 because of E.76). For the analytic spectrum see Theorem
5.29 (a).
(iii) The branched covering (C, 0) -> (M, 0) is branched along {(x, t)\x =
t2 = 0}. Compare Theorem 5.29 (c). The generating function F : С -*¦ С has
three different values on n^\t) forteM with t2 ф 0 because of E.76) and
the definition of _/). Therefore K. = B.

92 Singularities and Coxeter groups
(iv) Formula E.79) can be checked by calculation. The equivalence (a)
(P) follows. Corollary 3.8 (b) shows (a) «=*¦ (y) (one can also see (P) «=> (y)
directly).
(b) We start with coordinates (r,, t2, t3) for (M, p) as in the proof of Theorem
5.29 (a). The proofs of Theorem 5.29 (a) and Lemma 4.22 (b) give a unique
construction of data (Z, 0) = (C, 0) x (C3, 0), (C, 0) С (Z, 0) and
az = dfi + xdt2 + (b2(t2, b)x2 + b{{t2, t3)x + bo(t2, t3))dt3 E.80)
as in Corollary 4.21 for the germ (M, p) = (C3,0) of an F-manifold.
The set (C,0) = U,3=i(C,>0) is the union of 3 smooth varieties, which
project isomorphically to (C3,0), and is isomorphic to the product of (C2, 0)
and (C, 0) П ({(jc, t)\ti=t3= 0}, 0).
The components (C,, 0) can be numbered such that the intersection numbers
of the curves (Q, 0) П ({(x, t) \ r, = t3 = 0}, 0) are p2 - 1 for i = 1, 2 and
p3 — 1 for i = 1,3 and for i = 2,3. The numbers p2 and p3 are defined hereby
and satisfy pi > p3 > 2.
The 1-formaz is exact on Creg and can be integrated to a continuous function
F : (C, 0) -> (C, 0) with F\c, = h + ft for a unique function f; e C{t2, t3].
Then Q = {(*, 01 x = |f} and
Э/,
Э/
E.81)
We will refine (f\,t2, h) in several steps and change Z, C, az, /b /2, /3 ac-
accordingly, without explicit mentioning.
1st step: The coordinates (fj, t2, t3) can be chosen such that (C, 0) -»• (C3, 0)
is branched precisely over {f б С3 112 = 0}.
2nd step: The coordinate t\ can be changed such that /i = 0.
Then Ci = {{x, t) | л: = 0} and
С, П Q = |(х, 0 I x = 0 = M J = {{x, t)\x=Q,t2 = 0).
E.82)
Because of /i|c,nc,_= /ilc,nc, = 0, the functions /2 and f3 can be written
uniquely as /2 = ff • g, f3 = tf • g with ^2, p3 > 1, g, g e C{f2, t3] - t2 ¦
C{t2, t3]. Now E.82) shows Ц = tf'~l ¦ unit and p-t > 2. Therefore pt = pi
and g and g are units, with g@) # g@) if p2 = p3.
3rd step: The coordinate fe can be chosen such that f2 = f^.
E.81) yields b0 = 0 for i = 1 and fc, = -b2 ¦ p2 ¦ ff ~' for i = 2 and
1г 47 —'
5.5 3-dimensional and other F-manifolds
93
for (' = 3. The first, third and fifth factor on the right are units, therefore we
have ft = r
= rf~2
unit.
4th step: The coordinate t3 can be changed such that g = YlfLo 8i • 4 +
We have brought the germ (M, p) to a normal form as in (a). The numbering
of C\, C2, C3 was unique up to permutation of C\ and C2 if p2 > p3 and arbi-
arbitrary if p2 = p3. The choice of t2 was unique up to a unit root of order p2.
Everything else was unique. ?
Remark 5.31 Certain results of Givental motivate some expectations on the
moduli of germs of F-manifolds, which are satisfied in the case of Theorem
5.30.
An irreducible germ (M, p) of a massive F-manifold is determined by
its restricted Lagrange map (L(r), A,(r)) -+ (Г*М(г), T*M^) -+ (M<r), /A>)
(section 3.4). Suppose that (M, p) is 3-dimensional with TpM = Qm. Then
(L^r\ Я(г)) decomposes into a product of two Lagrange curves, a smooth one
and a plane curve (C, 0) (Theorem 5.29 (a), Proposition 3.7).
If we fix only the topological type of the curve (C, 0), we can divide the
moduli for the possible germs (M, p) into three pieces:
(i) moduli for the complex structure of the germ (C, 0),
(ii) moduli for the Lagrange structure of (C, 0),
(iii) moduli for the Lagrange fibration in the restricted Lagrange map.
Within the д-constant stratum SM = [q e M \TqM = QA)} of a representative
M, the moduli of types (i) and (ii) are not visible because the Lagrange structure
of the curve (C, 0) is constant along SM.
But the moduli for the Lagrange fibration are precisely reflected by 5M
because of a result of Givental [Gi2, proof of Theorem 3]: as a miniversal
Lagrange map, the restricted Lagrange map is stable with respect to small
changes of the Lagrange fibration which preserve the symplectic structure; that
means, the germ of the Lagrange map after such a small change is the restricted
Lagrange map of (M, q) for a point q e SM close to p.
In view of Theorem 4.15 and Theorem 5.29 (b) and (c) there is one module
of type (iii) if mult(C", 0) = 3 and no module of type (iii) if mult (C, 0) = 1
or 2.
Fixing the complex structure of the plane curve (C, 0), the choice of a
Lagrange structure is equivalent to the choice of a volume form. Equivalence
classes of it are locally parameterized by Яд/1)((С, 0)) ([Gi2, Theorem 1],
[Va5]), so the number of moduli of type (ii) is /л, — r (Theorem 3.5 (b)). It is

94
Singularities and Coxeter groups
equal to the number of moduli of right equivalence classes of function germs
/ : (C2, 0) -* (C, 0) with (/"'@), 0) = (C, 0).
The /a-constant stratum of a plane curve singularity in the semiuniversal un-
unfolding is smooth by a result of Wahl ([Wah], cf. also [Matt]) and its dimension
depends only on the topological type of the curve. So one may expect that the
number of moduli of types (i) and (ii) together depends only on the topological
type of (C, 0) and is equal to this dimension.
(But a canonical relation between the choice of a Lagrange structure and the
choice of a function germ for a plane curve (C, 0) is not known.)
In the case of Theorem 5.30 these expectations are met: the topological type
of the plane curve is given by the intersection numbers p2 — 1 and pi — 1;
the last one gP3-2 of the complex moduli is of type (iii), it is the module for
the ^-constant stratum and for the Lagrange fibration; the other рз — 2 moduli
(go, • • •. 8п-з) are °f types (i) and (ii). One can check with [Matt, 4.2.1]
that p2 — 2 is the dimension of the /u.-constant stratum for such a plane curve
singularity.
Finally, at least a few examples of germs (M, p) of F-manifolds with TqM =
<2B> will be presented.
Proposition 5.32 Consider M = C3 with coordinates (t\, t2, b) and T*M with
fibre coordinates у i, y2, уз- Choose p2, Рз € N>2. Then the variety
L = {(y, t) e T*M | yi = 1, У2(у2 - p2t?-]) = y2y3
1) = 0} E.84)
has three smooth components and is the analytic spectrum of the structure of a
simple F-manifold on M with TqM = gB). The field
Э 1 3 1 Э
E.85)
is an Euler field of weight 1.
Proof. One checks easily that a = y\dt\ + y2dt2 + уз&з is exact on the three
components of L, that the map &c '¦ 1м —* л+Ol is an isomorphism of Од/-
modules, and that E in E.85) is an Euler field.
The weights of E are positive. This shows via the Lyashko-Looijenga map
that (M, o, e) is a simple F-manifold (cf. the proof of Theorem 5.21 (a)). D
Remark 5.33 In [Gi2, Theorem 15] the restricted Lagrange maps of two other
series of simple F-manifolds with M = С and TqM not a Frobenius manifold
5.5 3-dimensional and other F-manifolds
95
are given, they are the series an (n > 3) and Я„ (и > 4) (also S| = Ab
32 = Hi, п2 = A2, ft3 = tf3).
They have Euler fields of weight 1 with positive weights. The analytic spectra
of 3,, and Q.n are isomorphic to С х ?„_|Bп - 1) and С2 х E,,_2B« - 3),
respectively. Here E^B^+1) is the open swallowtail, the subset of polynomials
in the set of polynomials [z2k+l + a2z2k~^ Л h a2k+\ \ a2,..., a2k+} s C}
which have a root of multiplicity > к + 1 ([Gi2, p. 3256]). It has embedding
dimension 2k.
The germs ((M, 0), o, e) are irreducible for 3„ and Я„, the socle АппГоМ(щ)
of TqM is the maximal ideal m С ТцМ itselfin the case of 3„ and has dimension
n — 2 in the case of Я„.
Givental [Gi2, Theorem 15] proved that the germs (M, 0) for 3„ and Я„
are the only irreducible germs of simple F-manifolds whose analytic spectra
are products of smooth germs and open swallowtails. Generating functions in
the sense of Definition 4.23 are due to O.P. Shcherbak and are given in [Gi2,
Proposition 12].

Part 2
Frobenius manifolds, GauB-Manin connections,
and moduli spaces for hypersurface singularities

Chapter 6
Introduction to part 2
The notion of a Frobenius manifold was introduced by Dubrovin in 1991 [Dul],
motivated by topological field theory. It has been studied since then by him,
Manin, Kontsevich, and many others. It plays a role in quantum cohomology
[Man2] and in mirror symmetry.
But the first big class of Frobenius manifolds had already been constructed
in 1983 in singularity theory. K. Saito [SK6][SK9] studied the semiuniversal
unfolding of an isolated hypersurface singularity and its GauB-Manin connec-
connection. He was interested in period maps and defined the primitive forms as volume
forms with very special properties in relation to the GauB-Manin connection.
Any primitive form provides the base space of a semiuniversal unfolding of
a singularity with the structure of a Frobenius manifold.
He proved the existence of primitive forms in special cases and M. Saito
proved their existence in the general case [SM2][SM3]. Using the work of
Malgrange [Mal3][Mal5] on deformations of microdifferential systems, M.
Saito showed that the choice of a certain filtration on the cohomology of the
Milnor fibre yields a primitive form and thus a Frobenius manifold.
This construction of Frobenius manifolds in singularity theory has been
quite inaccessible to nonspecialists, because the GauB-Manin systems are
treated using the natural, though sophisticated language of algebraic analysis
and especially Malgrange's results require microdifferential systems and cer-
certain Fourier-Laplace transforms. This also made it difficult to apply the
construction.
The first purpose of part 2 of this book is to give a detailed account of a sim-
simplified version of the construction. This version stays largely within the frame-
framework of meromorphic connections and is sufficiently explicit to work with it.
The second purpose is to present several applications. The most difficult one
is the construction of global moduli spaces for singularities in one д-homotopy
class as an analytic geometric quotient.
99

100
Introduction to part 2
Outlines of the construction and the applications are offered in the following
two sections. In these outlines the reader can jump straightaway to the main
points. An orientation is given how all the material in the subsequent chapters
is used and a motivation to study it.
The applications are given in chapters 12, 13, and 14. In chapter 12 we
give a canonical complex structure on the /г-constant stratum of a singularity
and an infinitesimal Torelli type result, which strengthens a result of M. Saito.
In chapter 13 the global moduli spaces for singularities are constructed and
symmetries of singularities are discussed, extending some work of Slodowy
and Wall. In chapter 14 the G-function of a Frobenius manifold is used to study
the variance of the spectral numbers of a singularity.
The construction of Frobenius manifolds in singularity theory is carried out in
chapter 11. It requires the majority of the results which are presented in chapters
7 to 10. In chapters 7 and 8 a lot of material on meromorphic connections is
given, most of which is known, but not presented in this form in the literature.
It will be used for the discussion of the meromorphic connections in chapter 9
and 10.
In chapter 10 most of the known results on the GauB-Manin connection for
(unfoldings of) singularities are put together in a concise survey. In chapter 9
Frobenius manifolds are defined and certain meromorphic connections, which
arise from them, are studied. This extends some work of K. Saito, Dubrovin,
and Manin.
Sabbah generalized most of K. Saito and M. Saito's construction to the case
of tame functions with isolated singularities on affine manifolds [Sab3][NS]
[Sab2][Sab4]. But the details are quite different; there one uses oscillating
integrals, and the results are not as complete as in the local case. The case
of tame functions is important for mirror symmetry. A special case had been
studied by Barannikov [Ba3]. All of this is discussed in sections 11.3 and 11.4.
6.1 Construction of Frobenius manifolds for singularities
A Frobenius manifold here is a complex manifold M with a multiplication о and
a metric g on the holomorphic tangent bundle TM and with two global vector
fields, the unit field e and the Euler field E. The multiplication is commutative
and associative on each tangent space, the metric is flat, and all the data satisfy
a number of natural compatibility conditions (see Definition 9.1).
Let / : (Cn+', 0) -» (C, 0) be a holomorphic function germ with an isolated
singularity at 0. The dimension of its Jacobi algebra 0c"+',o/(|f-. • • • > Щ-) ='¦
O/Jf is the Milnor number /л. A semiuniversal unfolding is a function germ
6.1 Construction of Frobenius manifolds for singularities 101
f ; (C"+1 x O\ 0) -»¦ (C, 0) with F|(C"+1 x {0}, 0) = / and such that the
derivatives |f |(C"+1 x {0}, 0), г = 1,..., ц, represent a basis of the Jacobi
algebra; hereV t) = (*„, ...,*„, f,,.... rM) e Cn+1 x C.
One can choose a representative F : X -» Д with Д = B\ С С, М = В? С
С and X = F-'(A)n(B"+1 х М) С С"+1хС for suitable small s, S, 9 > 0.
One should see it as a family of functions F, : X П (Ben+1 x {t}) -» Л, para-
parameterized by t 6 M, with Fo = /. The manifold M is the candidate for a
Frobenius manifold.
Its tangent bundle carries a canonical multiplication: The critical space
С С A" of the unfolding is defined by the ideal 7F = (Ц,.... f?). Its pro-
projection ргс,м ¦¦ С -> M is finite and flat of degree /г. There is a canonical
isomorphism
Э
—
3F
F.1)
of free Ом-modules of rank д, the Kodaira-Spencer map. Here TM is the
holomorphic tangent sheaf of M. The natural multiplication on the right hand
side induces a multiplication о on Тм- This was first observed by K. Saito
[SK6][SK9]. With e = a-'(llc) as the unit field and E := a~\F\c) as the
Euler field, M carries a canonical structure of an F-manifold (M, о, е, Е) with
Euler field (Definition 2.8 and Theorem 5.3). The Kodaira-Spencer map gives
for each tangent space T,M an isomorphism
]M,o,E\,)=( (?
\xeSin
(T,M
Jacobi algebra of (F(, x), mult., [Ft]). F.2)
In order to find a metric one may look for a similar isomorphism as in F.1)
from the tangent sheaf to a sheaf with a nondegenerate symmetric bilinear form.
Such a sheaf exists, the sheaf
пх+/А,М F-3)
of relative differential forms with respect to the map <p in F.5) is a free
Ом-module of rank \x and is equipped with the Grothendieck residue pairing
Jf : Qf x Qf -*¦ Ом (see section 10.4)
It is also a free (prc,jw)*C>c-module of rank 1; generators are represented by
suitable volume forms unit(x, f)djco... dxn. The choice of a generator induces
isomorphisms
(ргс,м)*Ос
F.4)

102
Introduction to part 2
and the pairing Jf yields a metric on T M. But one needs special volume forms,
the primitive forms of K. Saito, in order to obtain flat metrics and Frobenius
manifold structures on M. Their construction uses the GauB-Manin connection.
In the following we intend to give an idea of this construction. The details
can be found in section 11.1. The map
cp:X-+AxM, (x, t) ь» (F(x, t), t) F.5)
is a C°°-fibration of Milnor fibres outside of the discriminant V := <p(C) с
A x M. The cohomology bundle
H"= (J Hn(q>-\z,t),C) F-6)
(z.t)eAxM-i)
has rank /г and is flat. There is a distinguished extension 7i@) of its sheaf of
holomorphic sections over the discriminant.
The sheaf Ti^ can be defined in two ways. In terms close to the relative de
Rham cohomology it is
It is a coherent and even free Од хМ -module of rank \x [Gre]. It has a logarith-
logarithmic pole along V, that means, the derivatives of sections in 7i@) by logarithmic
vector fields are still in 7i@). The residue endomorphism along Vreg has eigen-
eigenvalues (^, 0,..., 0) and is (for n > 2) semisimple. The second description
of 7i@) is this: it is the maximal coherent extension of the sheaf of holomorphic
sections of H" over V with these properties along V (see Lemma 10.2 and
section 10.3).
One can extend the cohomology bundle H" uniquely to a flat bundle over
С x M - V and the sheaf 7i@) to С х М. Now the key point is to look for
extensions Wa) of 7i@) over P'xM which are free OPi x ^-modules and which
have a logarithmic pole along {oo} x M. Figure 6.1 may help to visualize the
situation.
log. pole
{oo} x M
A x {0}
log. pole_
Figure 6.1
6.1 Construction of Frobenius manifolds for singularities 103
Extensions with a logarithmic pole are not difficult to obtain. Denote by
H°° the д-dimensional space of the global flat manyvalued sections in
(C - A) x M of the extended cohomology bundle (there is only the monodromy
around {oo} x M). This space Я°° is equipped with a monodromy operator.
There is a one-to-one correspondence between locally free extensions W0)
with a logarithmic pole along {oo} x M and monodromy invariant (increasing
exhaustive) filtrations U, on H°° (see sections 7.3 and 8.2).
But for a free Opi xM-module H*® one needs special filtrations U..
First one observes that it is sufficient to show that the restrictions of the
sections to P1 x {0} yield a free OPi-module. A classical theorem (cf. for
example [Sab4,1 5.b], [Mal4, §4]) on families of vector bundles over P1 then
asserts that the sheaf over P' x M is a free module (for M sufficiently small).
The germs in (A x {0}, 0) of the sections in 7i@) form the Brieskorn lattice
Hq [Bri2], a free C{z}-module of rank /z. In general it does not have a logarith-
logarithmic pole at 0, but its sections have moderate growth, so it is regular singular.
Varchenko showed that the principal parts of its sections give rise to a Hodge
filtration on Я°° (see section 10.6). His construction was modified to obtain
Steenbrink's Hodge filtration F' on H°°. Here Я°° is canonically identified
with the space of the global flat multivalued sections on the cohomology bundle
over A* x {0}.
Now M. Saito found that an opposite filtration U. to this Hodge filtration
(see Definition 10.19) is what one needs. In fact, he did not look for extensions
to oo, but he constructed from U, a basis for Щ with properties such that he
could apply Malgrange's results. The fact that an opposite filtration U. gives
rise to an Opt -free extension to P1 of the Brieskorn lattice Щ with a logarithmic
pole at oo is a solution of a Riemann-Hilbert-Birkhoff problem and is discussed
in section 7.4.
The existence of opposite filtrations U. to F' follows from properties of
mixed Hodge structures. In general U. is not unique.
Now fix a choice of U. and the corresponding extension 7i@). Denote by
n : P1 x M —у M the projection. The sheaf я,7^@) of fibrewise global sections
is a free Ом -module of rank д. It contains the /x-dimensional subspace of
sections whose restrictions to {oo} x M are flat with respect to the residual
connection along {oo} x M (see section 8.2).
The residue endomorphism along {oo} x M acts on this space. It turns out
that it acts semisimple with eigenvalues — скь ..., —aM (the rational numbers
a\,..., aM are the spectral numbers of Щ, see sections 7.2 and 10.6). The
smallest spectral number or i has multiplicity 1 (Theorem 10.33).
Let v\ be a global section on 7i@) which is flat along {oo} x M with respect
to the residual connection there and which is an eigenvector with eigenvalue

104
Introduction to part 2
—a\ of the residue endomorphism. It is uniquely determined up to a scalar. It
turns out that it is a primitive form in the sense of K. Saito.
It yields a period map and an isomorphism л*Н@) = TM. The residual
connection along {00} x M induces a flat connection on TM. In order to see
that the isomorphism F.4) for Vi as volume form gives a flat metric on TM
one needs two things: K. Saito's higher residue pairings (see section 10.4) and
the fact that the Hodge filtration F* on H°° is part of a polarized mixed Hodge
structure. There is a polarizing form S on H°° which also has to be respected by
the opposite filtration U.. It is related to the higher residue pairings (see section
10.6). Altogether one obtains the following (see Theorem 11.1).
Theorem 6.1 A monodromy invariant opposite filtration U. to the Hodge fil-
filtration F' on #°° induces a flat metric g up to a scalar on M such that
(А/, о, е, Е, g) is a Frobenius manifold.
All the material in Sabbah's book [Sab4] and the second structure connections
in Manin's book [Man2, II§2] have been very helpful for carving the above
version of the construction of Frobenius manifolds in singularity theory.
The second structure connections in [Man2] are a family Vw, s e C, of
meromorphic connections over P1 x M for a semisimple Frobenius manifold.
The definition generalizes to arbitrary Frobenius manifolds (chapter 9). In the
case of singularities the connection V(~l' turns out to be isomorphic to the
(extended) GauB-Manin connection with the sheaf 7i@\ The connection V@)
was also defined by Dubrovin [Du3]. In the singularity case, K. Saito had already
defined the germs at 0 of all the connections V(i> in a different way [SK9, §5].
6.2 Moduli spaces and other applications
Singularities which are contained in a д-constant family have isomorphic
Milnor lattices and for и ф2 even the same topological type. They are called
/x-homotopic. Singularities which differ only by a local coordinate change are
called right equivalent and should be considered as isomorphic. One may ask
about the moduli space of right equivalence classes of singularities in one ц-
homotopy class.
Let us fix a /x-homotopy class ? с m2 с Ос+'.о of singularities and an
integer к > /x + 1. The right equivalence class of a singularity / € ? is deter-
determined by its &-jet jkf e m2/mi+1. The set jk? с m2/m*+1 is a quasiaffine
variety (possibly reducible as a variety, but connected as a topological space).
The algebraic group jk7l of ?-jets of coordinate changes acts on it. The quotient
parametrizes the right equivalence classes of singularities in the
6.2 Moduli spaces and other applications
105
/x-homotopy class ?. One knows that the orbits all have the same dimension,
but a priori not much more about the group action. We can prove the following
(see Theorem 13.15).
Theorem 6.2 The quotient ]к?/]кТ1 is an analytic geometric quotient.
A priori this is a global statement. But with the construction of unfoldings and
with some results of Gabrielov and Teissier one can translate it into statements
on semiuniversal unfoldings of singularities in ?. Then one can use the rich
structure of their base spaces as Frobenius manifolds.
Theorem 6.2 includes the claim that the quotient topology on jk?/jk1l is
Hausdorff. I now want to sketch the proof of that part in one page.
Consider a singularity / e ? and a semiuniversal unfolding F with base
space M as in section 6.1. The /x-constant stratum in M is S^ = {t € M | Sing
F, = {x}zndF,(x) = 0}.
A result of Scherk [Sche2] says that for any t e M the datum in F.2)
determines the right equivalence classes of the germs (F,, x) for x e Sing F,.
So the base space as F-manifold (M, о, е, Е) with Euler field knows the right
equivalence classes of all the singularities above it. There is a related result
of Arnold, Hormander, and others on Lagrange maps and generating families.
It implies (Theorem 5.6) that the germ ((M, t), о, е, Е) of an F-manifold for
t G 5M determines the germ of the unfolding at the singular point of F, up to
right equivalence of unfoldings.
Now consider for /, F, M, and 5Д(/) as above a sequence (r,-)ieN with
U e SM(/)andr,- -> Ofori -*¦ со and suppose that there is a second singularity
/ with F, M, 5Д(/) and (?¦); defined analogously and with a sequence of
coordinate changes <p, such that Fu — F?. о щ. One has to show that then /
and / are also right equivalent. This will imply that the quotient topology is
Hausdorff. Figure 6.2 illustrates the situation.
There is no possibility of controlling the coordinate changes <p, and finding a
limit coordinate change. But it turns out that they induce unique isomorphisms
<PiM ¦¦
, t,), о, е, Е) -»•
i), о, е, Е)
F.8)
of the germs of F-manifolds. With the construction of Frobenius manifolds we
can control these and show that there is a subsequence^which gets stationary for
large (' and extends to an isomorphism (M, 0) -> (M, 0). Then опекал apply
Scherk's or Arnold's and Hormander's result and see that / and f are right
equivalent.
Essential for controlling the sequence <pi<M is the strong link between flat
structures on M and filtrations on H°° by Theorem 6.1. The coordinate changes

106
Introduction to part 2
M Э
M D
Figure 6.2
<Pi also induce isomorphisms
<pUtoh : (Я00, Я2°°, Л, S, F'fo)) -+ (Я --, ^z-
here Я00 is the space from section 6.1 with canonical lattice Я^°, monodromy
operator h, polarizing form S and Hodge filtration F*(r,) from the singularity
Ft,-
The data (Я00, Я|°, Й, 5, F»(/)) encode polarized mixed Hodge structures.
There exists a classifying space DPMHS for them and a period map
Sn ->¦ DPMHS, t м- F\t). F.10)
Я~, /Г,
F.9)
The discrete group Gz = Aut(ff°°, Я^°, h, S) acts properly discontinuously
on DPMHS (because the mixed Hodge structures are polarized). This implies
that the isomorphisms <Pi,coh are all contained in a finite set. Via the construction
of flat structures on M the same then holds for the isomorphisms (pi,m (for more
details see section 13.3).
One can be more precise about the local structure of the moduli space
jke/jkil- The group of automorphisms Aut((M, 0), о, е, Е) is in fact finite
and acts on the /x-constant stratum 5M С М. Similar arguments as above yield
the following for a germ of the moduli space (see Theorem 13.15).
Theorem 6.3
Uk?/JkK, [jkf]) S (?„, 0)/Aut((M, 0), о, е, Е).
We can equip the /x-constant stratum with a canonical complex structure. An
opposite filtration U. as in Theorem 6.1 induces a flat metric up to a scalar on
6.2 Moduli spaces and other applications
107
U and a unique flat structure. By construction there exists a basis of flat vector
fields Si, .--.iSm with
The coefficients su e
in
F.П)
F.12)
are close to a part of Dubrovin's deformed flat coordinates (see section 11.2)
and they determine the //.-constant stratum (see section 12.1).
Theorem 6.4 The ^-constantstratum (Stf, 0) is the zero set of the ideal (e,j | aj —
1 — a.; < 0). This ideal is independent of the choice of Si,..., SM and even of
the choice of the opposite filtration U..
I expect that this ideal provides in general a nonreduced complex structure.
But it is very difficult to compute. I do not have examples. I also expect that the
induced complex structure on the moduli space jk^/jk^l is a good candidate for
a coarse moduli space with respect to some functor of /x-constant deformations
over arbitrary bases.
In [Hel][He2][He3] Torelli type questions for hypersurface singularities
were studied. There is a datum which is even finer than the polarized mixed
Hodge structure (Я°°, H$°, h, S, F*)of a singularity, the datum (H°°, H™, h,
S, Щ), the Brieskorn lattice together with topological information. A classify-
classifying space Dbl for such data was constructed in [He4]. It is a fibre bundle over
the classifying space DPMhs- The group Gz also acts properly discontinuously
on Dbl- There is a local period map
D
BL,
F.13)
for the /x-constant stratum of a singularity and a global period map
DBL/GZ F.14)
for a /x-homotopy class of singularities. A global Torelli type conjecture [He2]
asks whether F.14) is always injective. In all known examples it is true, but
a general answer is still unknown. Now Theorem 6.2 shows at least that the
moduli space on the left is an analytic variety. The map F.14) is now a morphism
between varieties.

108
Introduction to part 2
It is easy to see that F.13) is injective (for small 5M) if 5M is smooth. M. Saito
[SM4] used this fact to show that for general 5M it is finite-to-one. (He did not
have the classifying space Dbl, but considered a period map to a bigger space,
a subset of a certain flag manifold.) With the flat coordinates of a Frobenius
manifold structure on the base space M we can show a stronger infinitesimal
Torelli type result (see Theorem 12.8).
Theorem 6.5 The period map 5Д -*¦ Dbl is an embedding for small S^.
All these applications made use of the fiat structure on M which is induced
by an opposite filtration, but not of the metric itself and not much use of its other
properties, the multiplication invariance and the potentiality (see Definition 9.1).
These give rise to an extremely rich hidden structure on Frobenius manifolds
which has been uncovered by Dubrovin and Zhang [DuZl ] [DuZ2] and Givental
[Gi8]. Exploiting this for singularities is a big task for the future. We can present
one surprising application.
There is a function G{t) associated to each semisimple Frobenius manifold
([DuZl][DuZ2][Gi7], cf. section 14.2). Following a suggestion of Givental we
show (Theorem 14.6) that it extends in the singularity case to a holomorphic
function on the whole base space. It has the stunning property
48
=: y.
F.15)
The spectral numbers a,- satisfy the symmetry a,- + a^+i_,- = n — 1. So one
can consider <~ as their expectation value. Then - JX^a,- - ^J is their
variance. The G-function gives a grip at this variance. In the case of a quasiho-
mogeneous singularity /, one has / e J/ and E\o = 0 because of F.2). This
shows the following (Theorem 14.9).
Theorem 6.6 Iff is a quasihomogeneous singularity then у = 0. The variance
of the spectral numbers is
2 ) VI
It is part of the motivation for the following conjecture.
F.16)
Conjecture 6.7 For any isolated hypersurface singularity, the variance of the
spectral numbers satisfies
12
F.17)
Chapter 7
Connections over the punctured plane
The only initial datum in section 7.1 is a monodromy operator. For the corre-
corresponding flat vector bundle over the punctured plane C* notions such as the
elementary sections and the V-filtration are introduced. In section 7.2 Oc-free
extensions over 0 with regular singularity at 0 of the sheaf of holomorphic
sections of the vector bundle are discussed. Comparison with the V-filtration
leads to the spectral numbers and certain filtrations. Sections 7.1 and 7.2 are
elementary and classical.
The subject of section 7.3 is those extensions over 0 which not only have a
regular singularity at 0, but also a logarithmic pole. There is a correspondence
between such extensions and certain filtrations, which is not so well known. It
has a generalization in section 8.2. It is used in section 7.4 for the solution of
a Riemann-Hilbert-Birkhoff problem. This is based on ideas of M. Saito. It is
central to the construction of Frobenius manifolds in section 11.1.
In section 7.5 a formula is given for the sum of the spectral numbers in a
global situation, when one has on a compact Riemann surface a locally free
sheaf and a flat connection with several singularities as above. It is useful in the
case of P1 for the Riemann-Hilbert-Birkhoff problem.
7.1 Flat vector bundles on the punctured plane
Let us fix a holomorphic vector bundle H -*¦ C* of rank д > 1 with a flat
connection V on the punctured plane С* = С — {0}. We want to discuss
special sections in H and extensions of the sheaf H of its holomorphic sections
over 0 6 C.
Of course this is classical and has been done in many ways (e.g. [Del],
[AGV2], [SM3], [Hel], [Ku], [Sab4]). But we have to establish comfortable
notations in order to discuss later the information which is contained in certain
109

по
Connections over the punctured plane
extensions of H over 0, for example in the Brieskorn lattice (section 10.6). We
more or less follow [SM3][Hel].
A positively oriented loop around 0 induces a monodromy h on each fibre
Hz, z € C*, of the bundle. The monodromy determines the bundle uniquely up
to isomorphism. Let h = hs- hu = hu- hs be the decomposition into semisimple
and unipotent parts, N :— log hu the nilpotent part, and
:= кег((/ь - A.): Hz -+ Hz),
G.1)
the decomposition into generalized eigenspaces. We will use the universal
covering
e:C->C*, f н> e2jrlf. G.2)
Global flat sections A of the bundle e*H -у С induce via the projection pr :
e*H -*¦ H maps pr о А : С -*¦ H, which are called global flat multivalued
sections. The space of these global flat multivalued sections is denoted by H°°.
It is canonically isomorphic to each fibre (e*H)(, f e C. The monodromy h
acts on it with eigenspace decomposition H°° = 0Л Н?° and with
h A(f) = A(r + 1) * G.3)
for any Л б Н°°.
Now we can define some special global sections in H. Fix A e H™ and
a € С with enia - X. The map
С ^ H, fn e(af) exp(-f /V)A(f),
is invariant with respect to the shift f i-> f + 1 and therefore induces a holo-
morphic section
es(A,a) : C* -»¦ Я,
fore(f) = г G.4)
of the bundle #. It is called an elementary section [AGV2] and is usually
denoted informally as
/ N \
zaexp -logz- —• A,
V 2л-(/ n
G.5)
a is called its order. It is nowhere vanishing if А ф 0 because the twist with
e(af) exp(—f TV) is invertible.
The symbol Ca denotes the space of all elementary sections es(A, a), A e
H™, with a fixed order a. The map
фа : Ял°° -> С",
es(A, a)
G.6)
7. / Flat vector bundles on the punctured plane 111
is an isomorphism of vector spaces. By definition one has
z -es(A,a) = es(A, a + 1), z о т/га = фа+\, ' G.7)
(-N \
Va,es(A, a) = a ¦ es(A, a - 1) + es ( ^—7A, a - 1 I,
!,«),
/ N \ N
(zV8 - a)e5(A, a) = es --—7A, a I = --—r
\ Zjvi / Zni
G.8)
z • ca ->¦ Ca+1 bijective,
V3z: С -> С" bijective iffa^O,
zV3 - a = : Ca -»¦ С nilpotent,
2 2л i
so C" is a generalized eigenspace of zV8;. We simply call these spaces С
To obtain a filtration for these eigenspaces we fix a total order -< on the set
{a | e~2jtia eigenvalue of h] U Z, which satisfies
a -< a + 1,
a4|S iff a + 1 ч /3 + 1,
Va, ^Э/neZ a < P+m.
G.9)
Later h will be quasiunipotent and, if not said otherwise, the order will be the
natural order < on Q. But different orders can also be interesting (cf. Remarks
11.7). To simplify notations, we will write the usual symbol < for the order <.
It should be clear when the usual order < on E is meant and when the order -<
on [a | е~1л1а eigenvalue of h} U Z is meant.
From now on we will concentrate on germs at 0 of sections in H, that means,
on the stalk (i*H)o at 0 of the sheaf ?«H, where / : C* «-> С is the inclusion.
The space of function germs (i*Oc«)o and the operator Vaj act on this stalk. The
eigenspaces Ca and the elementary sections are identified with their images in
o. Then C" is characterized as
Ca = ker((zVa2 - a)
G.10)
for some m 3> 0. A basis of elementary sections in ©_i<a<o Ca induces а
basis of each fibre Hz, z e C*. Therefore
= ?ft (i.Oc.)oC" G.11)
is an (i,0c')o-vector space of dimension ti = dime Яг (г € С*).

112
Connections over the punctured plane
The space of all germs at 0 of sections of moderate growth (in the sense of
Deligne [Del, II §§1-4], comparing them with the flat multivalued sections) is
G.12)
The space V> °° is a C{z}[z ']-vector space of dimension ix. It is a regular
holonomic ?>c,o-module of meromorphic type [SM3, § 1]. Obviously it decom-
decomposes into T>c,o- and C{z} [z~' ]-submodules which correspond to a Jordan block
decomposition of H°°.
Within this space one has the Kashiwara-Malgrange V-filtration, an ex-
exhaustive decreasing filtration, indexed by {{a \ e"ia eigenvalue of h), <) and
defined by
C{z}Cfi.
One supplements this with
V>a :=
0
G.13)
G.14)
Then V* and V>ff are free C{z}-modules of rank д, satisfying
z : у" -» Va+1 bijective, G.15)
V3; : V
Va. : V
a » V"
>0
bijective if or > 0,
bijective,
Gr% := Va/ V>a = Ca canonically isomorphic. G.16)
Any section cd e V>-°° is a sum (often infinite) of unique elementary sections,
<w = ]T ¦$(«,«), s((o,a)eC, G.17)
a
whose orders are bounded from below by some number
or(«) := max(a | a> 6 V") = min(a | s(co, а) ф 0), G.18)
which is called the order of a>. The elementary section s(w, ct(co)) is the
principal part of со. All the sections s(co, a) are called the elementary parts
G.19)
Finally, we will need the ring
7.2 Lattices
113
of microdifferential operators with constant coefficients [Phi, part 2]. Just as
Clz} itIS a discrete valuation ring and a principal ideal domain. In view of the
action 3r'z* = шг*+' of 9^' on C{z}> the ring C{{9^'11 is designed to act
on C[z) such that C{z} is a free С{{Э-'}}-тос1и1е of rank 1 with generator 1.
It is well known ([Mal2, 4.1]) (and not hard to prove elementarily) that this
generalizes as follows.
Lemma7.1 ThemapVd. : C{z}Co+1 ->• C{z}Cafora i Z<0 is bijective, the
inverse extends to an action ofC{{d^}} such that C{z}Ca is a free C{{d~1}}-
module of rank dim C™.
Especially, V*~l and all V, V>a for a > -1 are free C[[d;l}}-modules
3
7.2 Lattices
We stay in the situation of section 7.1. Up to now the only initial datum was
the monodromy h. It determined the flat bundle H —>¦ C* up to isomorphy.
Everything in section 7.1 was developed from this.
But usually one has another ingredient, a C{z}-module in V>-o°, which
contains additional information (e.g. the Brieskorn lattice, cf. section 10.6).
One wants to understand this information by comparing with the V-filtration
and the elementary sections.
We first discuss C{z]-modules of У>~°0, most of the discussion also applies
to Qta-'H-submodules of V>.
A finitely generated C{z]-submodule of У >~°° is free, as C{z] is a principal
ideal domain. The name C{z] -lattice will be reserved for free C{z}-modules
of the maximal rank /x. A C{z]-lattice ?o can be extended uniquely to an
Ос-free subsheaf of rank ?i in i^H. The correspondence between C{z]-lattices
Co С V>~°° and Oc-free subsheaves of rank д of i^H whose sections have
moderate growth is one-to-one and justifies focusing on the stalks at 0.
A C{z}-lattice ?0 satisfies C{z][z~']?o = V>-oo,justasdotheC{z]-lattices
V". Therefore there exist a' and a" with Va' D C0D Va". The principal parts
of the sections of Co are placed together in the subspaces
= (V" П Co + V>e)/V>- С Or" = C.
G.20)
One can visualize them in figure 7.1.
Obviously zGry"' Cq = Gryz?o С Gry?o. The dimensions of the quotients
are
dim
1 Co) = dim Gr^?0 - dim Gi^' ?0 =: d(a) G.21)

114
Connections over the punctured plane
Са+Ъ
-1
V'
a + 1
¦>o
a +2
a + 3
Figure 7.1
and satisfy
G.22)
keZ
They give rise to the spectral numbers (<x\,..., a^) = Sp(Co) of the lattice Co
[AGV2], defined by
tt(/ | a,- = a) = d(a),
G.23)
and ordered by a; < • ¦ • < aM. Via the isomorphisms i/ra : H?° —> C"
(cf. G.6)), the subspaces Gry?o induce an increasing exhaustive uj-invariant
filtration F. on H°°,
for Я =
, -1 < a < О,
FPH°° :=
G.24)
G.25)
Remarks 7.2 (a) In the V-filtration, in ai < • • • < a^, and in —1 < a < 0 the
order -< from G.9) is used.
(b) In the case of the Brieskorn lattice of an isolated hypersurface singular-
singularity /(*o. ¦ ¦ ¦, xn), the filtration F* = Fn_. is essentially Varchenko's Hodge
filtration ([Val], cf. also section 10.6). It reflects the information contained in
the principal parts of the sections in the Brieskorn lattice and is already highly
trancendental. The question of how to treat the higher elementary parts of the
sections and the whole information in the Brieskorn lattice leads to M. Saito's
7.2 Lattices
115
work [SM3, §3], which will be taken up in section 7.4, and to my Torelli type
results ([Hel]-[He4] and section 12.2).
Lemma 7.3 A C{z}-lattice Co С V>~°° with spectral numbers a\ < • • • < aM
satisfies
Vе" dC0D V>a"-'. G.26)
Elements u>\,..., a>M e Co whose principal parts represent a basis of the space
1 Co form a C{z}-basis of Co-
Proof. We may suppose a(&>;) = a,-. The elements &>ь ..., &>д areC{z}-linearly
independent because in any linear combination with nonvanishing coefficients
at least one of the principal parts is not cancelled by anything. They generate a
C{z}-lattice?Q С Со- Because of G^C'q = Gr^?o for all a, one can enlarge the
order of elements of Co arbitrarily by adding elements of C'o. So Cq с C'o+V^
for some large ft and then Co = C'o.
In the same way one obtains У>а^ с С'о + Vp = C'o = Cq. The inclusion
V Э Co is obvious. ?
AfreeCKa-'H-modulero С V>- 'of maximal rank /miscalled a C{{d-1}}-
lattice. Spectral numbers ai,..., aM are defined as in G.21) and G.23). One
obtains an increasing exhaustive /ij-invariant filtration F"lg on H°° by
Ff*H? = if'1 v?Grv+p?o for X = e™, -1 < a < 0, G.27)
palg яоо _ д^ palg яоо (? 2g)
Lemma 7.3 holds analogously for C{{dz '^-lattices.
Often a subspace Co С V>-' is given which is a C{z}- and a C{{Зг"' }}-lattice.
Then the definitions G.21) and G.23) give the same set of spectral numbers
when one replaces z in G.21) by 3. But the filtrationsF. and Ffg may differ.
Lemma 7.4 (a) A C{z}-lattice Co С V>'1 is a C{{d-l}}-lattice if and only if
V-!?o С Co. A C{[d-1}} -lattice ?0 С V2" is a C{z}-lattice if and only if
zC0 С Co.
(b) The filiations F. and F?g of a C{z}- and C{{d-l])-lattice Co С V>~1
satisfy F",g = F_i =0 and are related for p > 0 by the formula
F°plgH? = П (-?j + к + a") FpH
where X = е'2*'", -1 < a < 0.
1.29)

116
Connections over the punctured plane
Г
7.3 Saturated lattices
117
Proof: (a) The inclusion ?Q Э V' for some a', Lemma 7.1, and Va l?0 С ?0
imply that ?0 is a C{{3~'}}-lattice, analogously when z?o С ?q for C{z}.
(b) This follows from the definitions of F. and F^lg and from the formula
2Va2 - a = -? on Ca (cf. G.8)). D
Remarks 7.5 (a) The filtration Fp g has the index 'alg' because in the case of
a hypersurface singularity f(xo,.... xn) the operator V8. has a more algebraic
flavour than the operator z (cf. sections 10.3 and 10.6).
(b) The Brieskorn lattice is a C{C~' }}-lattice, and F°'j. is the Hodge filtra-
filtration of Steenbrink [Stn] (and Scherk [SchSt], M. Saito [SMI], Pham [Ph3]).
Because of G.29) the filiations F°'l. and F,,_. coincide on the quotients of the
weight filtration (which is defined via N, cf. [Schm][AGV2] and section 10.5)
and are both Hodge filtrations of mixed Hodge structures. But F^J, behaves
better with respect to a polarizing form and is part of a polarized mixed Hodge
structure ([He4], cf. sections 10.5 and 10.6).
(c) This polarizing form is, after the monodromy, a second topological ingre-
ingredient and can be married to the structure {V~l, C, z, 9~') giving aC{{3~'}}-
sesquilinearformon V, which in fact coincides with the restriction to y>-1
of K. Saito's higher residue pairings and which fits together with the Brieskorn
lattice [He4]. This will be discussed in section 10.6.
7.3 Saturated lattices
We stay in the situation of section 7.1. A saturated lattice ?q is a free C{z}-
module ?q С (i*H)o of rank /z with z V3.,?o С ?q. Then the germs of sections
in ?o have moderate growth, that means, ?o С V>~°°. This follows from the
classical theorem of Sauvage, that the solutions of a system of linear differential
equations with simple pole have moderate growth (cf. for example [Del, II. 1]
or [Sab4, II.2]).
We will see that there are correspondences
saturated lattices
¦o- filtrations on H -*¦ C* by flat subbundles
-o- monodromy invariant filtrations on #°°
G.30)
and that the saturated lattices and these correspondences are invariant under
change of the coordinate z. They are even independent of the choice of the
order < in G.9). A result related to these correspondences has been given by
Sabbah[Sab4,IIIl.l].
The structure of a saturated lattice is summarized in the following lemma.
Lemma 7.6 (a) Let ?Q С V>~°° be a saturated lattice with spectral numbers
a,,.. •, aM (cf. G.23)j and filtration F. on H°° (cf. G.24) and G.25)). Then
G.31)
so ?o contains all the elementary parts s(co, a) of a section со s ?q. The spaces
?0ПСо С С and Fp С H°° are N-invariant and thus monodromy invariant.
(b) Any increasing exhaustive monodromy invariant filtration F. on Я°°
induces a saturated lattice in the following way.
The filtration F. on H°° induces a filtration of H -> C* by flat subbundles
FpH —> C*; the sheaf of holomorphic sections inthe subbundle FpH isdenoted
by FpH. Then
" G.32)
is a saturated lattice and F. on H°° is the filtration of?0 defined in G.24) and
G.25).
Proof, (a) Lemma 7.3 yields the inclusions Vе" D ?o Э V>a»-1. The oper-
operator zV3; acts on the space yei/y>ov~' = ®a,<ff<a(l_iCa, its generalized
eigenspaces are the spaces C". The subspace ?qIV>"ii~1 is invariant un-
under zV3j. This implies G.31) and zV8z(?0 П С") С^П С". The formula
z V8j - a = -?, on Ca (cf. G.8)) gives the ЛГ-invariance of ?0 П С and of
Fp С Я°°.
(b) Exhaustive means that there exist integers a < b with 0 = Fo С fj =
H°°. Then V>a D ?0 Э V>b~l. The stalk (i*FpHH is an («„OcOo-module
and invariant under Vs.. The C{z}-module <C{z\Ca+p is invariant under zV8t.
Therefore ?o is a saturated lattice. The equalities
Grv+P?o = ?оП C+p = (i,FpH)o П Ca+P G.33)
for -1 < a < 0 show that G.24) and G.25) give the filtration F. on H°°. О
If <p : (C, 0) -> (C, 0) is an isomorphism of germs then <p(z) itself can be
considered as a new coordinate on (C, 0). One can lift <p to an automorphism Ф
of the flat bundle H -»• C* (in a neighbourhood of 0) such that Ф : Hz -» Нф)
is an isomorphism obtained from the flat structure by some path from z to (p(z).
The isomorphism Ф is not unique, only unique up to a power of the monodromy.
The isomorphism Ф maps (i*H)o to itself and C{z}-lattices to С[z]-lattices.

118
Connections over the punctured plane
But Ф(С") does not usually coincide with Ca: for the definition of elementary
sections the choice of a coordinate z was crucial. Also many C{z)-lattices are
not invariant under Ф.
Lemma 7.7 Let <p and Ф be as above. Then the sets C{z}Ca and all saturated
lattices are invariant under Ф.
Proof: One can use G.5) to see that an elementary section is mapped to a section
(not necessarily elementary) with the same order. Thus Ф(С°) С C{z}Ca, and
Ф-' gives Ф(С{г}Са) = C{z}Ca.
Now formula G.32) shows that any saturated lattice is invariant under Ф,
because the flat subbundles are invariant by definition of Ф. D
The subspaces V" of the Kashiwara-Malgrange V-filtration are saturated
lattices. Their independence of coordinates is of course known. We could have
considered the spaces C{z]Ca as the subspaces of the V-filtration for the flat
eigenspace subbundle ket(hs - е~1жш) -> С* with eigenvalue e"l7tia.
Now, for a coordinate free reformulation of the previous discussion, let M
be a 1-dimensional manifold, q e M a point, i : M — {q} e-> M the inclusion,
and H -» M - {q} a flat vector bundle with sheaf Ti of holomorphic sections.
The OM,q-modules Cq С (itH)q of rank д (= rank of H -+ M - {q}) are
called Ом,ч-lattices. They are the germs in q of CM-locally free extensions С
of H over q. If z is a coordinate around q then the vector field zdz generates
the logarithmic vector fields in a neighbourhood of q (cf. section 8.2). We can
rewrite the above definition of a saturated lattice coordinate freely.
Definition 7.8 An (9M,9-lattice Cq С (i*H)q is saturated if and only if it is
invariant under the logarithmic vector fields along [q] С М (cf. section 8.2).
An CV-locally free extension CofH over q has a logarithmic pole at q if and
only if the lattice Lq is saturated.
The Lemmata 7.6 and 7.7 give the following.
Theorem 7.9 The saturated lattices correspond one-to-one to the increasing
exhaustive filtrations by flat subbundles of the restriction of H to some disc
around q.
7.3 Saturated lattices
119
A saturated lattice Cq С (i*H)o is equipped with a residue endomorphism
Res, = zVs.
Cq/zCq
Cq/zCq
G.34)
here z € Ом,ч is a coordinate with z(q) = 0. The residue Res(/ is independent
ofthe coordinate: if? is any other coordinate with ?(<?) = Ofhenz^ = u(z)zdz
with и eOj,,,,u@)= 1.
In fact, the vector space Cq/zCq can be identified with the fibre at q of a
vector bundle on M which extends H and has Cq as the space of germs of
holomorphic sections. Then the monodromy automorphism h of the bundle H
(in a neighbourhood of q) extends to this fibre and coincides there with e-2lr'Res«
[Del, II 1.17]. The endomorphism e-2l"Resi has the same eigenvalues as h,
but may have a simpler Jordan block structure. The following more precise
(and well known) statements will be useful later. We again fix a coordinate z,
identifying (M, q) and (C, 0) and having the spaces С of elementary sections
at our disposal.
Theorem 7.10 LetC0 С (itH)obe a saturated lattice and F.the corresponding
(mondromy invariant) filtration on H°°.
(a) The coordinate z induces an isomorphism
ААА^фОгРН00 G.35)
p
which identifies the actions 0fe-27riReso On the left hand side and of h on the
right hand side.
(b) The eigenvalues of the residue endomorphism Reso are the spectral
numbers cc\,..., aM of Co.
(c) The endomorphism Res0 is semisimple if and only if N(FP) С Fp-\ for
all p.
(d) The endomorphism e-27r'Res» has the same Jordan normal form as h if no
two spectral numbers differ by a nonzero integer (nonresonance condition).
Proof, (a) The coordinate z provides the spaces С of elementary sections.
Because of G.31) and G.24) this yields isomorphisms
— 1<а<0 р
G.36)

120
Connections over the punctured plane
On each subspace Ca the monodromy h acts as exp (-2л iz Va.) (cf. G.8)). This
shows (a).
(b) and (c) follow immediately from (a) and its proof.
(d) Under the nonresonance condition there exists for each eigenvalue X of
h an index p(X) with Gr?wHf = Щ0. Now (d) also follows from (a). Q
Example 7.11 Suppose that the residue endomorphism Res0 of a saturated
lattice ?0 С (i»W)o is semisimple with eigenvalues (s, 0,..., 0) for some
s €C.
If s e С - (Z - {0}) then the monodromy is semisimple with eigenvalues
(«Г2*", 1,.... 1) and the lattice is ?0 = C{z}C° ф C{z]Cs.
If 5 e Z - {0} there are two possibilities:
(a) Either the monodromy is the identity; then the saturated lattice can be any
one in a family parameterized by P C° for 5 < 0 and by VHom(C°, C) for
s >0.
(b) Or there is one 2x2 Jordan block; then the saturated lattice is unique; for
s > 0itisC{z}ker(W : C° -» C°)+Vs,fors < OitisC{z}zs-N(C0)+V°.
7.4 Riemann-Hilbert-Birkhoff problem
The most recent references in book form on the Riemann-Hilbert problem and
the Birkhoff problem are [AB] and [Sab4]. In chapter IV in [Sab4] Sabbah
discusses several versions of the problem. One unifying general version can be
stated as follows.
Hypotheses: E, С P1 and S2 С P1 are two disjoint finite sets, H ->
P1 - (Si U ?2) is a flat vector bundle of rank /x > 1 and with sheaf H
of holomorphic sections, and ? is a free 0pi-x;2-module of rank /x with an
isomorphism?|P! - (Si U S2) = H.
Problem: Does there exist an extension of ? to a free OV\ -module ? which
has logarithmic poles along the points of S2?
Because of Theorem 7.9 one can extend ? to a locally free CPi-module with
logarithmic poles along E2 without any problem. The requirement that ? shall
be a free СУ -module makes the problem difficult.
Often the problem is formulated in terms of the (trivial) vector bundles which
correspond to the sheaves ? and ?.
Usually one makes additional assumptions on the poles along S i. At least
one supposes that for q € S| the coefficients of the connection matrix with
respect to a basis of Cq are meromorphic. A much stronger assumption would
7.4 Riemann-Hilbert-Birkhoff problem
121
be that the sections in Cq, q e Sb are of bounded growth, that means, the
connection is regular singular there.
The classical Riemann-Hilbert problem is the case Si = 0, the classical
Birkhoff problem is the case S, = {0}, S2 = {oo}, with the assumption that
the connection matrix with respect to a basis of ?0 has a pole of order < 2.
Sabbah [Sab4] calls the case S i = {0}, S2 = {oo} without special assumptions
on ?o the Riemann-Hilbert-Birkhoff problem.
A particular case of it was treated implicitly by M. Saito [SM3, §3]. He gave
(implicitly) a correspondence between certain solutions and certain filtrations on
the flat bundle H —> C*. The purpose of this section is to resume and generalize
this correspondence. In the case of hypersurface singularities such filtrations
exist because of properties of mixed Hodge structures (cf. section 10.5).
Let us stick to the situation and notations in section 7.1. Again H -» C* is
a flat vector bundle of rank /x and with monodromy h — hs ¦ hu = hu ¦ hs,
N = loghu, and П, Я°°, С, фа, Va, V>a, СЦд;1}} as in section 7.1. Again
?0 с V>~°° will be a C{z}-lattice (of rank д), and ? is the Ос-free, extension
of H to 0 with germ ?0 at 0.
Because of Theorem 7.9 the extensions of ? to locally free СУ -modules ?
with a logarithmic pole at oo correspond one-to-one to the monodromy invariant
finite increasing filtrations U. on H°°. Theorems 7.16 and 7.17 will describe
distinguished extensions to free Opi -modules.
The objects in section 7.1 for the point oo instead of 0 will all be equipped
with a tilde: the coordinate ?= j, satisfying
d? dz
the spaces Ca = C~a, V>a С V С V>-°°
a,,..., а„ for a C{z}-lattice ?ю С V>0.
-zdz, G.37)
,7^H0, the spectral numbers
Example 7.12 Let H -*¦ C* be the trivial bundle of rank 2 with basis e\, e2 of
flat sections. The following is a 1-parameter family of free CPi -modules ?(r),
r €<C, extending H to P1,
?(r)
+ ze2)
• ze\.
G.38)
The lattice ?oo('-) = V is constant and saturated with spectral numbers
Cfi, a2) = (—1, -1). The lattice ?0(r) С V° is not constant and saturated
only for r = 0; even the spectral numbers jump, (ai,a2)@) = A,1) and
(ai, a2) = @, 2) for г ф 0. A C{z}-basis of ?0(r) for г ф 0 whose principal

122
Connections over the punctured plane
parts represent a basis of the space 0a Gr^.Co(r)/Gr^z.Co(r) (cf. Lemma 7.3)
is given by
C0{r) = C{z] ¦ {rex + zei) Ф C{z} ¦ (z2e2), гфО.
G.39)
It does not extend to a basis at r = 0.
In the example J2i &i + Hi a> — 0- Theorem 7.20 will show that this holds
in general. We will be interested in free Opi -modules with a stronger property,
which includes a,- = —arM+i_,-. We resume two definitions from [SM3, §3].
Definition 7.13 (a) Let Co С V>~°° be a C{z}-lattice. A /x-dimensional sub-
space W С Co such that the projection pr : W —>¦ Co/zCo is an isomorphism
is the image of a unique section i> : Co/zCo —> Co with prov — id. The space
W is called a good .Co/z-Co-section if the following two equivalent conditions
hold:
(i) The nitrations pr(V П W) and (V П?о + zCo)/zCo on C0/zC0 coincide,
(ii) The space W has a basis whose principal parts represent a basis of the space
(b)If?0 С V>-l isaCffa^Ml-lattice, good ^o/V^'A-sections are defined
analogously.
Remarks 7.14 ([SM3, §3]) A basis of the image Im v of any C-linear section
v : Co/zCo -»¦ Co of a C{z}-lattice ?0 С V>-°° is a C{z}-basis of Co, by the
lemma of Nakayama. Always pr(Va <1W)C (Va ПС0 + zC0)/zC0.
In Example 7.12, the space W = С • (re, + ze2) + С • zex С Coif) is not a
good ?o/z?o-section for г ф 0.
The interplay between two filtrations on a finite dimensional space is dis-
discussed in [De2]. If F. and U. are two increasing filtrations on a finite dimen-
dimensional vector space then one has canonical isomorphisms
9Г Fp П
П !/,_, + Fp_, П Uq)
This helps to explain the equivalences in the following definition (cf. [De2],
[SM3, §3]).
Definition 7.15 Two increasing filtrations F. and U. on a finite dimensional
vector space are called opposite to one another if the following three equivalent
conditions hold:
7.4 Riemann—Hilbert—Вirkhoffproblem
123
(i)
0 then p + q = 0.
(ii) The vector space splits into a direct sum 0p Fp П f/_p.
(iii) One has decompositions Fp = ®q<p 1
for all p.
¦'qip
Theorem 7.16 Let С be an Ос-free extension over 0 of the sheaf"H of sections
of the flat vector bundle H -> C* with Co С V>~°°. The filtration which Co
induces on H°° by G.25) is denoted by F..
There is a one-to-one correspondence between the two sets of data:
(i) Extensions of С to Op-free modules С which have a logarithmic pole at
oo and satisfy: the ix-dimensional space ?(P') of global sections is a good
Col zCo-section,
(ii) Monodromy invariant increasing exhaustive filtrations U. on Я°° such that
F.H™ and U.H°° are opposite and, for A. # 1, F.H™ and U.+lH™ are
opposite.
The spectral numbers (a\,..., aM) at 0 and (c?j,..., c?M) at oo of such an
extension С are related by a,- = —
Proof. Let U. be a filtration as in (ii). The following explicit construction
of a basis of global sections of С from U. is a key idea in [SM3, §3] (cf. also
[He4, chapter 5]). The space H°° decomposes into
H°° =
™ П Fp П U-p
HFpn ?/,_„.
G.40)
p x^i p
These subspaces lead to distinguished spaces of elementary sections
g«+p :=zP^a (ЯД,(„ П Fp П f/@ or ,)_„) С Са+Р G.41)
for — 1 < a < 0, p € Z, with the properties
C" = 0Z-pGa+p Э Gr^^o = фг'б^' = Ga ®zGr»rlCo, G.42)
p<0
NGa = (ZVa2 - a)Ga С
G.43)
p>0
Formula G.43) follows from the monodromy invariance of U. and from z Vg. —
a — — ^r on C" G.8). The purpose of the filtration U. is really the splitting in
G.42) of the filtration of С by Grf,z'?o-
One can choose a basis of elementary sections s{,..., s^ of фа G™ with
orders <x(sj) = a,-. They form a basis of Hz for each z € C*. They are principal
parts of germs in Co- The point now is that there exist unique germs u,- in Co

124
Connections over the punctured plane
-•a+l
a a + l
Figure 7.2
whose principal parts are s,- and whose higher elementary parts are contained
*n ©0>a, Фр>о Z~PGP+P for each i = 1,..., /n: starting with some sections
whose principal parts are j,-, one can inductively for increasing i and increasing
orders erase all higher elementary parts in 0^>a. @p<qZ~pG^+p by adding
elements of Co. The uniqueness of the section
e CoП U
V
P>o
G.44)
is clear because the difference of two such sections would have an impossible
principal part.
Figure 7.2 shall illustrate the construction. The eigenspaces Ca should be
imagined as columns above the as; the picture does not take into account the
different dimensions of the С and the discreteness of the values a.
A space G& is nonzero if and only if /J 6 {a!, ..., aM}. Therefore the sections
Vi are sums of finitely many elementary parts, so they are global sections in
TC(C). They form a basis of Hz for each z 6 C* because the s,- do and G.44)
gives a triangular coefficient matrix.
The lattice at oo of the Opi -module С := ф; Opt ¦ и,- is
?°o = фС{г) • Vi = 0С|г)л = 0C{?)Ga. G.45)
i i a
Formula G.43) shows that it is saturated. Going through G.24) and Lemma 7.6
(b) one checks that the filtration on H°° which corresponds to C^ via G.30)
is precisely U.. Here one needs zfe = -zdz and Ca — C~a G.37). Obviously,
the spectral numbers C?!,..., aM) at infinity are a,- = —aM+i_,-.
It remains to go from (i) to (ii).
Let ? be as in (i). One can choose a basis vi,..., i>M of global sections
in ДР1) whose principal parts ^i,...,sM represent a basis of the space
7.4 Riemann-Hilbert-Birkhoffproblem 125
Co and have orders a,. Then these principal parts j|,..., $д
also form a global basis ofthe bundle Я -+ С*, because theyformaClzHz]-
basis of V>~°° and because they are elementary sections.
They generate а С (?)-lattice C'^ at infinity whose spectral numbers are
—a\, • • •, —oCf,.. But the lattice Сж is saturated and contains all elementary
parts of the u, (Lemma 7.6 (a)). Therefore C'^ С ?оо- If ?<» were bigger than
C'^, the spectral numbers a\,..., aM of C^would satisfy ^a,- + Ylai < 0-
But Theorem 7.20 will show
Therefore C'^ = Cx. Now one defines
¦Si CC01.
G.46)
G.47)
The identity G.42) holds. The identity G.43) holds because C^ is saturated by
assumption. Reading G.41) and G.40) backwards one obtains a monodromy
invariant increasing exhaustive filtration U. on H°° which satisfies the prop-
properties in (ii) and which is just the filtration corresponding to С,ж via G.30).
?
In the singularity case one is interested in C{z}-lattices ?oCP which
also are C{{3~' }}-lattices. Then a stronger result holds.
Theorem 7.17 Let С be an Ос-free extension over 0 ofthe sheaf H of sections
of the flat vector bundle H —у С* with Но С V>~1, and suppose that Co also
is a C[{d-1}}-lattice. The filiations which ?0 induces on H°° by G.25) and
G.28) are denoted by F. and F?lg.
(a) The two conditions for a monodromy invariant increasing exhaustive
filtration U. on H°° are equivalent:
(i) The filiations F. and U. are opposite in H^°; the filiations F. and U.+\
are opposite in H^forX ф 1.
(ii) The filiations F?s and U. are opposite in H™; the filiations F°'« and
U.+\ are opposite in H%° for Хф\.
(b) There is a one-to-one correspondence between the two sets of data:
(i) Extensions of С to Opt -free modules С which have a logarithmic pole at
oo and which satisfy: the [i-dimensional space ?(P') of global sections is
a good Со/У^1 Co-section.
(ii) Filtrations as in (a) (i) and (ii).

126
Connections over the punctured plane
Then ?(P]) is also a good ?o/zjCo-section; a basis of it is a C{z}- and C{{ 37 '}}-
basis of Cq. The spectral numbers at 0 and oo satisfy a,- — — aM+i_,-.
Proof, (a) We will prove that condition (i) for Hf° implies condition (ii) for
Hf°. Everything else is analogous.
We have Ff'f = F_, =0 because of ?0 С V>~1. For any p > 0 the
automorphism<pp := Пы^Ш +k">of Я1°° in <7-29) maPs fph\° t0 ^'*#Г
and respects U.Hf°. Therefore on Я,°°
<pp(Fp П f/_p) с Fp П [/_„, G.48)
(F, П U.p) © С/-Р-, = U-p = <pp(Fp П СЛ.„) Ф C/-P-i, G.49)
Я,°° = 0^а>П [/_„). G-50)
p
The equality dim Fp = dim Fp 8 now implies
(Bu-4)> G.51)
П U.p =
П f/_p
G.52)
(b) First we go from (ii) to (i). Suppose U, satisfies the properties in (a).
Theorem 7.16 and its proof yield an extension С and global sections v,- with
principal parts s,- and spaces Ga = фа;=а С - st с С. It remains to show
:0 = Ga ф Vr'Giy-'jCo- We need the formula
n
V^№"«n y-«
G.53)
for — 1 < a < 0, p > 0, ^ e Z. This follows from the monodromy invariance
of C/_? and from
z=
(—?) = ¦
Ca+l fora>-l G.54)
with (a + 1 — j^() being invertible.
The formulas G.41), G.53), and the definitions of F, and F°lg show
Ga+" = (Grav+P?o) П
П
П
G.55)
for — 1 < a < 0, p > 0. This formula together with the analogon of G.40) for
Fp shows that G.42) is also valid for a > -1 if one replaces z by V^~',
С = 0 VlGa+p D
v?Ga+p.
G.56)
p<0
T
7.4 Riemann-Hilbert-Birkhoff problem
127
This implies that ?(P') is a good ?o/V^'?
It remains to go from (i) to (ii).
Let? be as in(i)and vi,..., fM be a basis of global sections in ?(P') whose
principal parts s\,... ,sfi at 0 represent a basis of the space фа Gr" Ln/V~x
Gr°"'?o- It is sufficient to show that they also represent a basis of 0a
GfyLo/zGfy^Zo. Then one can apply Theorem 7.16.
We may choose v\,..., uM such that c*i,..., aM are their orders. Define
Ga:=
Ji CC".
Then
Ca = 0 VapGa+') fora > -1.
G.57)
G.58)
The lattice ?00 is saturated and contains with u,- also its principal part 57 (Lemma
7.6 (a)). Because of dz = —l,2dj. then
?»Э 0 0VsP;Ga+"D 0 C". G.59)
— l<a<0p>0 —l<a<0
The sum in the middle generates a CfzHattice ?'„, whose spectrum is al-
already —ot),..., — aM. Theorem 7.20 implies in the same way as in the proof of
Theorem 7.16 that ?<*, = C'^. Now dz = z" ¦ (—z'S?) again shows
Z^ n C° = Ga ф (zZoo П Ca) for a > -1 G.60)
and then
z^a t 1 -\ —p/-a
— v 1J
pel,
Gr»?o = G"®zGt<v-
Now one can apply Theorem 7.16.
+p fora
lC0 fora
>-l,
> -1.
G.61)
G.62)
?
Remarks 7.18 (a) The main result Theorem 3.6 in [SM3, §3] can be stated as
follows: Under the same assumptions as in Theorem 7.17 there is a one-to-one
correspondence between filtrations [/, as in Theorem 7.17 (a) (ii) and good
' W С Со with
Z W С W
G.63)
Comparing the proofs of Theorem 7.17 and [SM3, Theorem 3.6], one finds that
these /г-dimensional spaces W are precisely the spaces ?(P') of global sections
of extensions С as in Theorem 7.17 (b) (i). But this is quite nontrivial.

128
Connections over the punctured plane
M. Saito does not consider the questions of whether a basis of germs in W
is extendable to a basis of global sections in the bundle H -*¦ C* and whether
they give a logarithmic pole at oo. He does not formulate Theorems 7.17 and
7.16 and does not use the filtration F..
(b) The main application of Theorem 7.17 and of [SM3, Theorem 3.6] is the
same: to find a distinguished basis of sections in the GauB-Manin connection of
a versal unfolding of a singularity which induces a Frobenius manifold structure
on the base.
In section 11.1 we will use Theorem 7.17 and Theorem 8.7 in order to extend
the GauB-Manin connection in an explicit way to oo. M. Saito uses G.63)
in order to apply a result of Malgrange on deformations of microdifferential
systems ([Mal3, §5], [Mal5]). In the proof of Malgrange's result there also
is an extension of a connection along oo, but other ingredients are a Fourier
transformation and microlocal aspects.
(c) One can separate the proof of [SM3, Theorem 3.6] into four pieces:
(i) to construct from a filtration U. as in Theorem 7.17 (a) (ii) a good
?o/V^'?0-section W С ?o ([SM3, Proposition 3.4], similar to the first
part in the proof of Theorem 7.16 with V3"' instead of z),
(ii) to show that this section satisfies G.63) (a discussion of principal parts,
this is not so difficult),
(Hi) to recover a filtration U. from a good ?o/V;j"'^-section W с ?o with
G.63) ([SM3, Proposition 3.5]),
(iv) to show that this section is the unique section constructed from U. in (i)
(this is the most difficult piece; it is not even clear a priori that the elements
of a section W with G.63) consist of finitely many elementary sections).
(d) We will meet in the singularity case an extension of G.63) to the GauB-
Manin connection of a versal unfolding, cf. chapter 11, A1.20) and A1.54).
The coefficients will carry precious structure.
7.5 Spectral numbers globally
The results in this section are inspired by and partly due to [Sab4, IV 1.10].
We stay in the situation of section 7.1. The spectral numbers (a i,..., aM) of
a C{z}-lattice ?0 С V>'°° had been defined by
tt(j | a,- = a) = d(a) := dim Gr^?0 - dimGr?"'?0, G.64)
cf. G.21) and G.23), and were ordered by ai < ... < aM.
The V-filtration is independent of the coordinate z in С (Lemma 7.7), so the
spectral numbers of a lattice are also coordinate independent. But in G.9) a total
7.5 Spectral numbers globally
129
order < had been chosen. A different order may give a different V-filtration
and other spectral numbers. An example due to M. Saito [SM3, D.4)] will be
presented in Remark 11.7.
The following result shows that the sum ?, a,- is even independent of the
order <.
The determinant bundle det H = Ялд —> С* is a line bundle and carries an
induced flat connection. The monodromy is given by its eigenvalue det h e C*.
The determinant sheaf ?лм of an Cc-free extension ? to 0 of the sheaf H of
sections of H -*¦ C* is an Oc-free extension of the sheaf HAIM of sections of
Ялм -> С*. If ?0 С V*'00 then the sections of ?лд also are of bounded
growth at 0. More precisely, one has the following.
Theorem 7.19 Let С be as above with ?o С V>~°° and with spectral numbers
a\,...,a^caO. Then the germ C^ of C^ is a saturated lattice in 0»НЛДH
with residue eigenvalue Yl?=i ai-
Proof: One chooses a C{z}-basis a>\,... ,соц, of ?o whose principal parts
s!,..., sM represent a basis of @a Gr^ ?0/zGry~' ?0 and have orders a i,..., ад.
Then ft>i,..., а>ц generates ?q^ over C{z). It is sufficient to show
a>\ Л ... Л (Op = u(z)si Л ...
G.65)
with u(z) e C{z), "@) = 1.ThenC^ = C{z}«i л ... Aa>,l= C{z}si л ...
л Sfj. has residue eigenvalue ^ or,- (in the rank 1 case a lattice with sections of
bounded growth is automatically saturated).
The principal parts sit..., 5^ form a CMIz'^-basis of V>~°°, so
with unique af) e C, which satisfy au = 1 and
af? ф 0 only for 0", k) = (i, 0) or a, < aj + k. G.67)
Expanding a>\ Л ... Л ш^ with G.66) one has to see that all combinations of
summands except for the combinations of the principal parts are contained in
zC{z}s\ A... a s^. This is elementary; it follows from G.67) and the properties
in G.9) of the transitive order <. ?
Theorem 7.20 Let the following be given: a holomorphic vector bundle L -*¦
M of rank jx on a compact Riemann surface M with sheaf С of holomorphic
sections, afinite setT, С М, aflatconnectiononL\M--? such that the sections of
? have bounded growth everywhere. Let {a \ (q),..., <Хц(д)) denote the spectral

130
Connections over the punctured plane
numbers of the lattice Cq for q e T,. The line bundle L^ is the determinant
bundle ofL. Then
= -deg I*
G.68)
Proof. The restriction L^\M_-L is equipped with the induced flat connection.
The sheaf ?лд has logarithmic poles with residue eigenvalues ?,. at{q) at all
points q <=Y,, because of Theorem 7.19.
One can choose any nonvanishing meromorphic section on LA/X with zeros
and poles in a set ?' with ?' П ? = 0. It yields a meromorphic connection
1 -form w with simple poles in X/ U ?. The sum of its residues vanishes.
. G.69)
П
Chapter 8
Meromorphic connections
Section 8.1 is a reminder of logarithmic vector fields and differential forms
and some other classical facts. In sections 8.2-8.4 extensions with logarithmic
poles along a divisor V С M of the sheaf of holomorphic sections of a flat
vector bundle on M — V are discussed. In the case of a smooth divisor V in
section 3.2, there are three important tools for working with such extensions:
the correspondence to certain filtrations, the classical residue endomorphism
along V, and the (less familiar) residual connections along V, whose definitions
require the choice of a transversal coordinate.
Extensions to singular divisors V are treated in section 8.3 in greater gener-
generality than in the literature. If an (automatically locally free) extension to Vreg
with logarithmic pole is given, then there exists a unique maximal coherent
extension to V. It is locally free only under special circumstances. The case of
a normal crossing divisor is discussed in section 8.3, the GauB-Manin connec-
connections for singularities provide other very instructive examples (Theorem 10.3,
Theorem 10.7 (b)). In Section 8.4 only some remarks on regular singularities are
made.
8.1 Logarithmic vector fields and differential forms
For the reader's convenience we put together some definitions and results from
[SK4][Del][Ser], which will be useful in sections 8.2 and 8.3.
Let M be an m-dimensional complex manifold, V С М a hypersurface, and
g : M —> С a holomorphic function such that V = g~'@) and the ideal sheaf
(g) С Ом is everywhere reduced. A (locally defined) holomorphic vector field
X e TM is logarithmic if X g e (g).
A (locally defined) meromorphic g-form со with poles at most on V is log-
logarithmic if ga> and g dtw are holomorphic g-forms. These definitions are obvi-
obviously independent of the choice of the defining function g for V. The sheaves of
131

132
Meromorphic connections
all logarithmic vector fields and differential <7-forms are denoted Der^ (log P)
and ?2^(logP), respectively.
Example 8.1 (Smooth divisor P.) Suppose M С Сш and g = zi- Then these
sheaves are free CV-modules,
^©ffio^-dz,.
Z\
i=2
= /\SlM(\ogV).
The logarithmic vector fields are the vector fields tangent to P.
(8.2)
(8.3)
(8.4)
This case of a smooth divisor P is essential for the general case because of
some codimension 2 arguments. The singular part Vsing С М of the divisor
P has codimension > 2 in M. Let i : M - Ря„г «^ М denote the inclu-
inclusion. Then i*OM-vsl4 = Ом by the Riemann extension theorem, and also
i*((g)\M-v^) = (g). The above definitions show
I = *-*(DerM(logP)|M_psiJ, (8.5)
i = i*(uqM(\ogV)\M-Vting). (8.6)
So the logarithmic vector fields are the vector fields tangent to Preg. Formula
(8.4) also holds for general M and P.
Example 8.2 ([Del, II §3], the normal crossing case.) Suppose M = Cm and
g = z\ ¦ Zi ¦. ..-Zk f°r some к with 1 < к < m,soP = (J*_, С' x{0}xCm~'.
Then the sheaves Der^(logP) and S^(logP) are free OM-modules,
Э
Zi3z7
(8.7)
(8.8)
i=k+l
and the sheaves S^(logP) for 9 > 1 are determined by (8.4). Here equality
for the restrictions to M - Vsing follows from Example 8.1. The sheaves on
the right are free Ow-modules. Then equality follows from (8.5), (8.6) and
8.1 Logarithmic vector fields and differential forms
133
The sheaves of logarithmic vector fields and differential forms are coherent.
This is clear for DerM(logP) because that is the kernel of the map TM ->¦
Ом Kg)* Xh Xg. But it is not so clear for the differential forms. It will
follow from (8.9), but first we will see it in a way which will also be useful in
section 8.3. The proof uses the following result of Serre.
Theorem 8.3 [Ser, Theorem 1 and Proposition 7] Let X be a normal variety,
Y С X a subvariety of codimension > 2, and T an Ox-y-coherent sheaf on
X — Y without torsion. Let i : X - Y ^ X denote the inclusion.
(a) Then i*T is Ox-coherent if and only if there exists some Ox-coherent
extension ofT.
(b) (i) lfT is reflexive and i*T is Ox-coherent then i*T is reflexive and it is
the only reflexive extension ofT.
(ii) Especially, if there exists some locally free extension ofT then i+T is this
locally free extension.
A sheaf is reflexive if the map to its bidual is an isomorphism. The statement
in Theorem 8.3 (b) (ii) is elementary and was already used in Example 8.2.
Arguments of the following type can be found in [Del, II 5.7] and [Mal7].
They will again be used in section 8.3.
Theorem 8.4 Let M be a manifold and P a hypersurface. The sheaves of
logarithmic vector fields and differential forms are coherent and reflexive
Ом-modules.
Proof. Following Hironaka, there exists a resolution / : (M, P) —*¦ (M, P),
that is, a manifold M, a normal crossing divisor P С М, and a proper holomor-
phicmap/: M ->¦ M with F~\V) = V and such that /: M-P-> M-V
and / : M - f'^H M- Vsing are biholomorphic.
The sheaf Der^(log D) on M is locally free because of Example 8.2. By
Grauert's coherence theorem for proper maps (cf. for example [Fi, 1.17] the
sheaf /*Der^(log P) is OM -coherent. It is a coherent extension to M of Der^
(\ogT>)\M-vsing- By Theorem 8.3 (a) and (8.5), then DerM(logP) is coher-
coherent and reflexive. The same applies to the sheaves of logarithmic differential
forms. ?
Reflexiveness of Derw (log P) and ?2lM (log P) also follows from the fact that
these sheaves are dual OM -modules by the inner product
DerM(logP) x
OM, (X, со) v+ co(X)
(8.9)

134
Meromorphic connections
([SK4, A.6) and A.7)]). This duality is a consequence of Example 8.1 and (8.5),
(8.6).
In general, DerM(log?>) and fi},(log?>) are not necessarily locally free OM-
modules. If they are, then PcMis called a free divisor. Several criteria for
a divisor being free are given in [SK4]. In the case of singularities or, more
generally, massive F-manifolds with Euler fields, the discriminants are always
free divisors, often the bifurcation diagrams (section 4.3).
8.2 Logarithmic pole along a smooth divisor
Let M, V, g be as in section 8.1, M a complex manifold, V с М ahypersurface,
and g : M -» С holomorphic with V = g~'@) and everywhere reduced ideal
sheaf(s)c0M.
Let H -> M — V be a flat vector bundle with connection V and sheaf H of
holomorphic sections.
Deligne [Del] gave a precise meaning to the notion regular singular for
(meromorphic) equivalence classes of Ow-coherentextensionsofW and showed
that there is always precisely one regular singular equivalence class of exten-
extensions. We will come back to this in section 8.4
K. Saito [SK2] emphasized that one should look not just at this equivalence
class of extensions, but at specific extensions of TL which are usually given
naturally. Here and in section 8.3 CV-coherent extensions with logarithmic
poles will be discussed; the case of a smooth divisor here, the general case in
section 8.3.
An important fact and tool is that in the case of a smooth divisor V the
correspondence G.30) between extensions with logarithmic poles and some
filtrations extends to the higher dimensional case. Two other tools in this case
are the residue endomorphism and a family of residual connections.
Definition 8.5 Let M, V, g be as above and С be an CV-coherent extension of
H to M. The pair (?, V) (or simply ?, if the connection is unambiguous) has
a pole of order < r + 1 (or of Poincari rank r) along V for some r e Z>0 if
1 ,
V : С ->• — SUlog?>)®?., (8.10)
gr
The pair (?, V) has a logarithmic pole along V if r = 0.
Remarks 8.6 (a) We are interested here only in the case r = 0.
(b) This definition is standard at least in the case r = 0, C, V arbitrary and
in the case r arbitrary, С locally free, V smooth. In the second case it is usually
formulated in terms of vector bundles [Sab4][Man2].
8.2 Logarithmic pole along a smooth divisor
135
(c) The definition is independent of the choice of g.
(d) Because of the duality (8.9) between logarithmic vector fields and
1-forms, (?, V) has a logarithmic pole if and only if Vx maps С to itself
for any logarithmic vector field X.
Let(M, ?>)beisomorphicto(Am,{0}x Д'"-'), where Д = {z eC\\z\ < 1}
С С is the unit disc, and let g : M -> С be holomorphic with V = g~] @)
and (g) С OM reduced. As above, H -> M — V is a fiat vector bundle of rank
\i with connection V and sheaf H of fiat sections; i : M - V -> M is the
inclusion. The aim of this section is to make the following statements precise
and explain them.
Theorem 8.7 (a) Any OM-coherent extension С of ТС to M which has a loga-
logarithmic pole along V is a locally free О м -module.
(b) There is a one-to-one correspondence between sheaves С on M as in
(a) and increasing exhaustive flltrations of H -» M — V by flat subbundles
FPH -+ M -V, pel.
(c) There is a unique О-^-Ипеаг residue endomorphism Res on the locally
free Ov-module (C/g ¦ C)\v- It is independent of g.
(d) The function g induces aflat residual connection Vrej'« on (C/gC)\v- The
residue endomorphism Res is flat with respect to this residual connection.
Two functions g and g with 'g/g e O*M induce the same residual connection
if(g/g)\v = constant.
Remarks 8.8 (i) Often one considers a priori only extensions of the bundle
U _>. м — x> to a vector bundle on M with a logarithmic pole along V. This is
justified by (a). But in the case of a nonsmooth divisor V this approach would
be too narrow.
(ii) Some particular filtrations of H -> M - V are the filtrations which are
trivial on each subbundle of generalized eigenspaces of the monodromy, that
means, 0=Fp_i С Fp = total subbundle for some p 6 Z. They correspond to
the sheaves С as in (a) whose residue endomorphisms satisfy the nonresonance
condition: no two eigenvalues of Res differ by a nonzero integer.
That these extensions С are uniquely determined by the eigenvalues of their
residue endomorphisms is classical ([Del, II §5], [Manl]).
(iii) A point of view which together with Theorem 7.9 gives Theorem 8.7(b)
is formulated in [Mal4, p. 404] (cf. also [Sab4, III 1.20]): A locally free O^x[Q)-
module?'on Дх@)сД" = М which is an extension of the sheaf of sections
of Я|д.Х@) with a logarithmic pole along {0} can be extended uniquely to a
sheaf С on M as in (a).

136
Meromorphic connections
(iv) The definition and the uniqueness of the residue endomorphism Res are
classical. Res and V"g are presented together in [Man2, II 2.1] and [Sab4,
0.14.(b)], except for the precise dependence of Ves's on g.
(v) Although V№* depends on g and is thus less canonical than Res, it will be
extremely useful in the case of bundles on M = P1 x M' where the coordinate
z on P1 or i serve as g.
First we choose coordinates and set M = Am, V — {0} x Am~l, g = z\ ¦ The
dependence on the choices will be discussed later.
We have to repeat the construction in section 7.1 of the elementary sections
with respect to z\. Again h = hs ¦ hu = hu ¦ hs denotes the monodromy of the
bundle H ->• M -V, with semisimple part hs, unipotent part hu, and nilpotent
part N ~ loghu. The fibres Hz, z € M — V decompose into generalized eig-
enspaces, Hz = фл Ягд with ЯгД = ker(/is — X).
These subspaces form the flat subbundles H^ = {JzeM_x> ^г,х wim sheaves
of holomorphic sections. Again
е:Нх Д"-1 -+M-V, (?, z2, •. •, zm) н+ (e2*1^ ,Z2,. ¦., z») (8.11)
is a universal covering, and the space
H°° := {pr о A : H x Am~l -> H | A is a global flat section of е*Я} (8.12)
is the /^.-dimensional space of global multivalued flat sections. Here pr:
e*H -*¦ H is the projection. The monodromy acts on it, H°° = 0X H?° with
Ях°° = ker(fts - X).
Then for any A 6 H?° and aeC with e~2jria = X there is an elementary
section
es(A,a):M-V-* H,
z i-> ехрBлгга?) exp(-^)A(f, z2, • ¦ •, Zm)
(8.13)
here e(^, Z2 zm) — z. Formulas analogous to G.7) hold, especially
(N \ N
--—A, a I = --—es(A, a), (8.14)
2jt( J 2ni
, a) = 0 for/>2.
(8.15)
Again C" denotes the space of all elementary sections es(A, a), A 6 H?°, with
a fixed order a. A basis of С is a basis of sections of the subbundle H^.
8.2 Logarithmic pole along a smooth divisor
137
Now let FPH -*¦ M — V, p e Z, be flat subbundles which form an increas-
increasing exhaustive filtration. We construct from this filtration an OM -locally free
extension LoiH with a logarithmic pole along V in two ways; the first way is
more explicit; the independence of choices is easier to see in the second way.
1st way: The filtration F, H induces a monodromy invariant filtration F.Я00.
One may choose a basis Aj e FpU)H™jy г = 1, ¦ ¦ •, M, of H°° which splits
this filtration and the eigenspace decomposition. Then
С :=
• es(Ah u(J)
(8.16)
7=1
= Mj), —1 < Re(aO')) < 0. Obviously these elementary sections
form a basis of the bundle H -*¦ M — V. The extension С of H to M has loga-
logarithmic pole along V because of (8.14), (8.15) and the monodromy invariance
of F.tf00.
2nd way: Let FpH^) denote the sheaf of holomorphic sections of FPH^)-
L := 0 E *'* W» n Ou ¦ Ca+P (8.17)
X pel
for e-2*ia = X, — 1 < Re(a) < 0, and i : M ~ V ^-> M the inclusion. One
sees that (8.16) and (8.17) give the same extension ?.
The residue endomorphism for this extension L is the Op-linear map
Res = ziVa : LjzxL\v
(8.18)
on the locally free Op-module C/ziQv of rank д.
The residual connection VreStZl on ?/z\?\v is the flat connection whose flat
sections are generated by the classes in L/ziC\v of the elementary sections
es(Aj, a( j) + p( j)) from (8.16). Obviously Res is Vre"' -flat. The eigenvalues
of the residue endomorphism are the numbers a(j) + p(j), j = 1,..., /x.
Theorem 8.7 follows from the next lemma.
Lemma 8.9 (a) The construction in (8.16) and (8.17) gives all Ом-coherent
extensions CofH with a logarithmic pole along V.
(b) Let a\,..., а^Ье an Ом-basis of an extension L as in (8.16) and (8.17)
and Ai(z)^- + X^i>2 ¦Ai(z)dz,- be the connection matrix with respect to it.
Then Ai@, Z2, • ¦ ¦, zm) and 5Z/>2 ^'@. Zi, ¦¦¦, zm)dz,- are the matrices for
the residue endomorphism Res and the residual connection Vres'*' with respect
to the basis g\\v, . ¦ ¦, ctm|d of Ljz\L\v-
(c) The residue endomorphism Res is independent of the choice of coordi-
coordinates. The residual connection Ves>Zl depends only on z\, not on Z2, ¦ ¦ ¦, zm,
andWKS'z> = Ves'?l if(z\/z\)\v = constant.

138
Meromorphic connections
(d) The OM-locallyfree extension OM ¦ С с иКщ ofHw depends only on
a, not on (zi, ..., zm)- It is the unique extension with logarithmic pole along
V and with a as the only eigenvalue of its residue endomorphism.
(e) The construction of С in (8.17) from the filtration F.H is independent
of the choice of coordinates. It gives a one-to-one correspondence between
increasing exhaustive filiations of H -> M -V by flat subbundles FpH ->
M - V and OM-coherent extensions ofH with a logarithmic pole along V.
Proof, (a) Let ? be an 0M -coherent extension of H to M with a logarithmic pole
along V. The induced extension to A x {0} of the sheaf of sections of H \ A, x {0,
is ?' := ?/fe? H h zm?)Ux@!- It is a locally free OA xH) -module of rank
/x and has a logarithmic pole in {0} С Л х {0}.
The 0(AxfO),o)-lattice C'o is saturated. By Theorem 7.9, ?' and ?(, come
from an increasing exhaustive filtration of Я|Дх@) by flat subbundles. The flat
extensions to M - V of these subbundles are denoted by FpH -> M - V,
peZ.
They induce by (8.16) and (8.17) an OM-locally free extension ? of H.
Choose an OM-b&sis es(Aj, a(j) + p(j)) of ? as in (8.16). Their restrictions
to A x {0} form an 0Ax{o}-basis of ?(,. The sections es(Aj, a(j) + p(j)) also
form an i*OM-v-basis of iji.
Any germ a € ?0 is a unique linear combination
. es(Aj< a{
withal e (i*0M-v)o- We claim au) e OMfi:
Any derivative of a by j^,..., ¦?- is also contained in ?0 because it has a
logarithmic pole along V; its restriction to A x {0} is contained in C'o. So the
restriction to A x {0} of any derivative of a(j) by -f-,..., ^- is in ОдХ{0).
This shows o-W e 0м,о and ?0 С ?0. Now any germs "ox,..., стд е ?0
with о-;|(дХ{о),о) = «(А,, а(У) + рО'))|(дХ(о),о) generate ?0, because the co-
coefficient matrix expressing O\,..., crM by the elementary sections is invertible
with holomorphic entries. Thus ?0 = ?o.
(b) This holds for any basis of ?0, because it holds for a basis of elementary
sections as in (8.16). For such a basis, the connection matrix is A{ ^ with A\
a constant matrix.
(c) A change of coordinates does not change A i @, zi, ¦ ¦ ¦, zm) and Res. The
definition of the elementary sections depends only on zi and they determine
V"-Zl. One sees easily that a change of zi to a coordinate z\ with (zi /zi)\v =
constant does not change ?/>2 A,@, z2,..., zm)dz,-.
8.3 Logarithmic pole along any divisor
139
(d) The corresponding statements for the restriction to A x {0} hold because
of section 7.3. One can use this and (a). But of course, the uniqueness of the
extension of i/H.(X) to an OM -locally free sheaf with logarithmic pole along
V and a as the only eigenvalue of the residue endomorphism is classical
([Del, II §5], [Manl], cf. Remark 8.8 (ii)).
(e) The first statement follows from (d), the rest with (a). ?
8.3 Logarithmic pole along any divisor
Let M, V, g be as in section 8.1, M a complex manifold, V a hypersurface, and
g : M -> С holomorphic with V = g~'@)and(g) С ОМ reduced everywhere.
Let H -> M - V be a flat vector bundle of rank д with sheaf of holomorphic
sections H.
Section 8.2 gives a correspondence between certain filtrations and coherent
extensions to M — Vsing of H with logarithmic poles along V - Vsing. To make
this precise one can choose some Riemannian metric on M and the correspond-
corresponding distance function d : M x M -*¦ R>q. For sufficiently small e > 0 and
S > 0 with S <K e the submanifold
M5,e := {z € M | d(z, V) < S, d(z, Vsing) > e) (8.19)
is diffeomorphic to a disc bundle over V - {z € V \ d(z,T>sing)<s},
having the same number of components as Vreg. Theorem 8.7 implies the
following.
Lemma 8.10 Any OM-v,hs-coherent extension of H to M - Vsing with a
logarithmic pole along V - Vsing is locally free. There is a one-to-one corre-
correspondence between such extensions and increasing exhaustive flltrations by flat
subbundles of the restriction of H to Mg:S — V.
The aim of this section is to study extensions to Vsing of sheaves as in Lemma
8.10. The main result is Theorem 8.11. It w ill be proved at the end of the section.
Let j : M — Vsing ^Mbe the inclusion.
Theorem 8.11 Let С be a locally free extension ofHtoM- Vsing with a
logarithmic pole along V — Vsing.
Then 7*? is OM-coherent and has a logarithmic pole along V. It is the only
reflexive extension of С
Example 8.12 Consider M = C2, g = zizz, V = g~\0), H = С х (M-D)
with the trivial flat connection. Each ideal (zf', Zj2) for h ,k2 e Z>0 is invariant

140
Meromorphic connections
under zi3Zl and цдц- So each such ideal is an Ом-coherent extension of
Om\m-v with a logarithmic pole along V. All these extensions coincide on
M — VSing. The only reflexive and even locally free extension under them is
OM. The last two statements illustrate Theorem 8.3 (b).
Example 8.13 Consider M = C3, g = Z1Z2Z3, V = g~'@), H = С2 х
(M — V) with the trivial flat connection and global basis e\, e2 of flat sections.
The sheaf
С := OM ¦
+ ?>м
+ Ом ¦ Z\Zi(e\ + е2)
is coherent and has a logarithmic pole along V. The restriction to M — {0} is
locally free. The sheaf С satisfies
? = MC\m-v,J (8.20)
and is therefore reflexive (Theorem 8.3 (b)), but it is not locally free.
Let us consider now the normal crossing case, M = Am, g = zi Zk,
V = g"'@) = LM=i DU) = U;-i z7'@); here A = {z e С | \z\ < I).
The flat bundle Я -+ M -V has к commuting monodromies Ло) = a?/) -A^,
у = 1,... Д, for the standard loops. A universal covering is
e :
x A
m~*
M - T>,
i...., ft,
(8.21)
The space of multivalued flat sections H°° is defined as usual,
H°° = {pr о A : H* x Am~k -> Я | A is a global flat section of e* Я}. (8.22)
All monodromies act on it. The indices of the simultaneous generalized eigen-
space decomposition H°° = фл Н?° can simply be considered as tuples X =
(XA),..., X(k)) of eigenvalues for the monodromies hA\ ..., A№).
The notion of elementary sections from sections 7.1 and 8.2 generalizes: if
one chooses A e Я?° and a = (aA),..., a№)) w4ith e-2jria0) = xU) then the
natural generalization of the formulas G.5) and (8.13) yields an elementary
section es(A,a).
Lemma 8.14 will show that any Cw-coherent extension of H with logarith-
logarithmic pole along V is generated by elementary sections. It is essentially due to
Deligne, but was first formulated in [EV, Appendix C]. We need the following
generalization of nitrations on H°°: a Z* -filtration (Pi \^k consists of subspaces
8.3 Logarithmic pole along any divisor
141
Pi С Я00 which are invariant with respect to all monodromies AA),..., h(k)
and which satisfy for a suitable m > 0
Pi =0 if lj < -m for some j, (8.23)
Pi = H°° iflj > m for all j, (8.24)
PL С PL if lj < I) for all j. (8.25)
Then Pi = фх Р;,х- A Z^-filtration (Pi) induces к increasing exhaustive
monodromy invariant nitrations F.(;) on Я°° by
F^ := U P,. (8.26)
For X^ an eigenvalue of /гУ) let a^} be defined by e~^iaW = ),<¦» and -1 <
Lemma 8.14 [EV, Appendix C] Fix a Zk-filtration
(a) For any l_ e ([—m, m] П Ъ)к and any X, choose generators АЦ of Р;,л.
The sheaf
(8.27)
these I X
w an Ом-coherent extension ofH with logarithmic pole along V.
(b) Fix a Z* -filtration PL and С as in (8.27). The filiations F^ on H°°
correspond together to one filtration on Ms,e — D (formula (8.19)j by flat
subbundles. The OM-vsin -coherent extension of Л which corresponds to this
filtration in Lemma 8.10 is C\M-vsing-
(c) Formula (8.27) yields a one-one correspondence between Ък-filtrations
and Ом-coherent extensions ofTi with logarithmic pole along V.
(d) The sheafX as in formula (8.27) is reflexive if and only if
for all I.
(8.28)
Then С = J*(C\m-vs,,is)-
(e) The sheaf С as in formula (8.27) is a locally free OM-module if and only
if (8.28) holds and the filtrations F^j) have a common splitting.
(f) If С as in formula (8.27) is reflexive then it is locally free outside of the
intersection of at least three components D(;).
Proof, (a) and (b) follow from the definition of elementary sections,
(c) See [EV, Appendix C] for the proof of this.

142
Meromorphic connections
(d) The sheaf ? is 0w-coherent, and C\M-vJII:, is locally free. By The-
Theorem 8.3 ? is reflexive if and only if ? = Л(?|м_р ). Then it must be
generated by the elementary sections which correspond to the Zk -filtration in
(8.28).
(e) The direction 'if is clear. For 'only if one can compare a generating
system of elementary sections as in (8.27) and an 0M,o-basis of ?0. It is not
hard to see that one can replace the basis elements by certain elements of
the generating set. This 0M,o-basis of elementary sections yields a common
splitting of the filtrations F.(j).
(f) Any two filtrations admit a common splitting. Q
Remarks 8.15 (a) A special case of Lemma 8.14 (e) is the case when the
filtrations on the simultaneous generalized eigenspaces H?° are all of the type
0 = Fp«.x.)-i c FpU,x) = Яоо for some numbers p(it k) ? z>
When these numbers p(i, k) depend only on i and A<;) then the residue
endomorphisms on the components D(i) satisfy the nonresonance condition
(Remark 8.8 (ii)): no two eigenvalues of one residue endomorphism differ by
a nonzero integer.
The Ом-locally free extension of H to M for the case when the real parts of
all eigenvalues of all residue endomorphisms are contained in [0, 1) was called
the canonical extension by Deligne [Del, II 5.5].
(b) The main arguments in the proofs of Theorem 8.4 and Theorem 8.11 are
due to Deligne [Del, II 5.7].
(c) In both proofs one could use [KK, Proposition С1.2 (ii)] instead of Serre's
result Theorem 8.3.
Proof of Theorem 8.11: One chooses a resolution / : (M, V) ->• (M, V) as
in the proof of Theorem 8.4. The first step is to extend the lift f*C to those
components of Vrcg which are mapped by / to lower dimensional subsets in
M. This can be done by choosing locally around these components for example
the filtration 0 = F~l С F° = (whole local flat subbundle) and applying
Lemma 8.10.
Then Lemma 8.14 (b) and (d) yields an O^-coherent extension to M. Just
as in the proof of Theorem 8.4, the direct image sheaf on M is an OM-coherent
extension of ? to M by Grauert's proper mapping theorem. Serre's result
Theorem 8.3 shows that j*C itself is ©^-coherent and reflexive and the unique
reflexive extension of ? to M. It has a logarithmic pole along V because ? has
a logarithmic pole along V - Vsing: The sheaf ? and therefore also j*? are
invariant with respect to logarithmic vector fields. D
8.4 Remarks on regular singular connections
143
8.4 Remarks on regular singular connections
Let M, V, g be as in section 8.1, namely M a complex manifold, V с М a
hypersurface, and g : M ->• С holomorphic with V = g~l@) and the ideal
sheaf (g) С OM everywhere reduced. Let i : M - V -> M denote the in-
inclusion. Let H —>¦ M — T> be a flat vector bundle with sheaf of holomorphic
sections H.
Deligne denned the notion regular singular not for a single CV-coherent
extension of "H, but for equivalence classes of such extensions with respect to
a meromorphic equivalence relation ([Del, II2.12]).
This equivalence class determines and is determined by a coherent OM[*]-
module in j, W which extends H (cf. forexample [Mal6][Mal7]). Here OM [*] =
OM[g~'] is the coherent ring sheaf of holomorphic functions onM-D which
are meromorphic along V.
Deligne showed that there is a unique OM[*]-coherent extension of H whose
sections have moderate growth with respect to the multivalued flat sections (for
the result and for the notion of moderate growth see [Del, II 5.7, 4.2, 4.1,
2.17,2.10,... ]. One says that this extension is regular singular along V, or the
connection is regular singular with respect to this extension.
Following K. Saito [SK2, E.1)], one can also say that an Ow-coherent exten-
extension of H has a regular singularity along V if it is contained in this meromorphic
regular singular extension.
In the 1-dimensional case M = Д = {z 6 С | \z\ < 1), 2? = {0}, the regu-
regular singular extension is the OM[*]-submodule of i*H with germ V>~00 at 0
([Del, II §1]).
In [Del, II 4.1] several criteria for regular singular are given. OM-coherent
extensions with a logarithmic pole along V were used only in the normal cross-
crossing case. In view of Theorem 8.11 one has the following.
Theorem 8.16 Let ?mewm be an OM [*]-coherentsubsheafofi*Tt which extends
TC. The following conditions are equivalent:
(i) The sheaf Cmewm is the regular singular extension, that means, its sections
have moderate growth with respect to the multivalued flat sections,
(ii) The sheaf Cmewm contains one (and then all) Ом -coherent extensions) of
"H with a logarithmic pole along V.
(in) The restriction of ?mem" to an open set U С М which intersects each
component ofVreg contains one (and then all) Ом -coherent extensions)
ofH\u-v with a logarithmic pole along V.

144
Memmorphic connections
Of course, this is well known, (e.g. [Lo2, 8.10] and references there). It
was the starting point for the definition of regular holonomic differential and
microdifferential systems ([KK][Bj]). There the meromorphic regular singular
extension is called a Deligne-type module (or D-type module).
Property (iii) says that regular singular is a codimension 2 property: it is
sufficient to check it outside a codimension 2 subset.
Chapter 9
Frobenius manifolds and second structure
connections
The definition and elementary properties of a Frobenius manifold M are put
together in section 9.1. Sections 9.2, 9.4, and 9.5 are devoted to their second
structure connections. These are connections overP1 x M on the lifted tangent
bundle of M with logarithmic poles along certain hypersurfaces. They come
from some twists of the original flat structure by the multiplication and the
Euler field. To know them is very instructive for the construction of Frobenius
manifolds for singularities, because in that case one of them turns out to be
isomorphic to an extension of the GauB-Manin connection.
Sections 9.2, 9.4, and 9.5 build on the definition and discussion of the sec-
second structure connections in [Man2] for the case of semisimple Frobenius
manifolds, on results in [Du3], and on [SK9, §5], where they together with
many properties had been established much earlier implicitly in the case of
singularities.
The second structure connections have some counterparts, the first structure
connections, which are better known. The latter are partly Fourier duals. The
main purpose of their treatment in section 9.3 (and in section 9.4) is to compare
them with the second structure connections.
9.1 Definition of Frobenius manifolds
Frobenius manifolds were defined by Dubrovin [Dul][Du3]. We follow the
notations in Manin's book [Man2, chapters I and II].
Here all manifolds will be complex. In the following, M denotes a manifold
with dim M = m > 1 (in the singularity case m = /z). A {k, Z)-tensor is an
О и -linear map T : T$k -» T$l. Here TM is the holomorphic tangent sheaf,
OM the structure sheaf.
A metric g is a symmetric nondegenerate B, 0)-tensor, a multiplication on
the holomorphic tangent bundle TM is a commutative (i.e. symmetric) and
associative B, 1) tensor.
145

146
Frobenius manifolds and second structure connections
The Lie derivative Lie^T of a (k, /)-tensor along a vector field is again a
(k, /)-tensor, as well as the covariant derivative Vx T with respect to a connection
V on M. Then Vr is a (k + 1, /)-tensor (cf. Remarks 2.7 and 2.13). The Levi-
Civita connection V of a metric g is the unique connection which respects the
metric, Vg = 0, and is torsion free, VXY — VYX = [X, Y] for local vector
fields X, Y.
The following definition is a bit more restrictive than in [Man2, chapter I]
because of the unit field and the Euler field.
Definition 9.1 A Frobenius manifold is a tuple (M, o, e, E, g) where M is a
manifold of dimension m > 1 with metric g and multiplication о on the tangent
bundle, e is a global unit field and E is another global vector field, subject to
the following conditions:
A) the metric is multiplication invariant, g(X о У, Z) = g(X, Y о Z),
B) (potentiality) the C, l)-tensor Vo is symmetric (here V is the Levi-Civita
connection of the metric),
C) the metric g is flat,
D) the unit field e is flat, Ve = 0,
E) the Euler field E satisfies Lie?(o) = 1 • о and Lie^(g) = D ¦ g for some
DeC.
Remarks 9.2 (a) For (M, o, e, g) as in the definition with condition A) (not
necessarily B)-E)) the C, 0)-tensor A defined by A(X, Y, Z) = g(X о Y, Z)
is symmetric. It arises from the B, l)-tensor о by contraction with the B, 0)-
tensor g. Because of Vg = 0 and the symmetry of g, the potentiality B) is
equivalent to the symmetry of the D, 0)-tensor VA.
In Theorem 2.15 it is shown that the potentiality is also equivalent to the
closedness of the 1-form s := g(e,.) together with the condition
Lie*oy(o) = X о Liey(o) + Y о Liex(o)
(9.1)
for local vector fields X,Y eTM.
(b) A manifold (M, o, e) with multiplication о and unit field e and (9.1) is
called F-manifold (Definition 2.8 and [HM][Man2]). Formula (9.1) for X =
Y = e gives Liee(o) = 0 • o. In this context a vector field E is already called
Euler field (of the F-manifold) if it satisfies Lie?(o) = 1 • o. This implies
[e,E)=e.
(c) The flatness of e, Ve = 0, is equivalent to Lise(g) = 0 and the closedness
of the 1-form s = g(e,.) (Lemma 2.16). Because the potentiality implies the
closedness of ? (cf. (a)), one could replace D) in the definition by Liee(g) = 0.
9.1 Definition of Frobenius manifolds 147
(d) The potentiality B) written out for arbitrary local fields X, Y, Z is
VX(Y о Z) - Y о VX(Z) - VY(X о Z) + X о Vy(Z) - [X, Y] о Z = 0. (9.2)
This formula can already be found in K. Saito's papers (e.g. [SK9, C.3.2)]). For
flat vector fields it is equivalent to the symmetry of VX(Y о Z) in X, Y, Z and to
the symmetry of Xg(YoZ, W)'mX, Y, Z, W (cf. (a)). Therefore it is equivalent
to the local existence of a potential F e OM,P with XYZ(F) - g(X о Y, Z)
for flat fields X, Y, Z.
(e) The property Lie?(g) = D ¦ g written out for local vector fields X, Y is
E g(X, Y) - g([E, XI Y) - g(X, [E, Y]) - D g(X, Y) = 0. (9.3)
Comparison with Vg = 0 shows that the A, l)-tensor
is skewsymmetric,
g(V(X), Y) + g(X, VA0) = 0.
(9.4)
The property Lie? (g) = D -g means that ? is a sum of an infinitesimal rotation, a
dilation and a constant shift. Therefore V maps a flat vector field X to a flat vector
field V(X) = [X, E] - yX. It is an endomorphism of the local system, in other
words, it is a flat A, l)-tensor, VV = 0. The eigenvalues of V? = V + f are
called the spectrum of the Frobenius manifold and are denoted by d\,.. ¦, dm.
They are symmetric around у and one of them is 1 because of [e, E] = e. We
order them such that dt + dm+\-i = D,d\ = 1.
(f) If V is semisimple then locally there are flat coordinates h,..., tm such
that the flat fields St = ^- are eigenvectors of V with
Then
(9.5)
(9.6)
for some г,- е С
(g) The multiplication with the Euler field is denoted by
U:TM-+TM, X (-> E о X.
(9.7)

148 Frobenius manifolds and second structure connections
9.2 Second structure connections
Let us fix a Frobenius manifold (M, о, е, E, g) and consider the lift
pr*TM —> TM
1 I
P1 x M -^ M
(9.8)
of the tangent bundle to P1 x M. The canonical lifts to pr*TM of the
tensors g, o, V, U will be denoted by the same letters. The connection V lifts
and extends to a flat connection on pr*TM such that V^Y = 0 for Y e
pr~lTM. Here г is a coordinate on С С P' and dz the vector field with Эг z = 1,
д,рг-[Ом=0.
With the multiplication and the Euler field one can twist this connection on
pr*TM in essentially two distinct ways. One obtains two series of flat connec-
connections, parameterized by one parameter s e С The first structure connections
V^s\ s e C, are meromorphic along {0} x M U {oo} x M. The second structure
connections V(s), s 6 С are meromorphic along V U {oo} x M, where
V - {(г, t)\U- zid is not invertible on T,M}.
(9.9)
The first structure connections V(j) are due to Dubrovin [Du3] (Lecture 3).
Some of the first and second structure connections are related by some Fourier-
Laplace transformations. Here we will concentrate on the second structure
connections. We will only give the definition and make some remarks on the
first structure connections in section 9.3.
Dubrovin considered also V@) and called it the GauB-Manin connection of
the Frobenius manifold [Du3, Appendix G]. In fact, we will see in section 11.1
that in the case of a hypersurface singularity f(xo,..., xn) the restriction of
V(~ ^ to С x M is isomorphic to the GauB-Manin connection on the cohomology
bundle of a semiuniversal unfolding of the singularity and that V@) and V(~^
are isomorphic if n is even and if the intersection form is nondegenerate.
The whole series V(J), s e C, was defined by Manin and Merkulov [MaM]
[Man2,111.2 and 1.4] for the case of semisimple multiplication. Their definition
also works in the general case and is given below in Definition 9.3.
But in the case of hypersurface singularities, K. Saito perceived this series
of connections much earlier [SK9, §5]: A primitive form for the GauB-Manin
connection (i.e. essentially the connection V'~2^) gives rise to a period map,
which has not so nice properties in the case of a degenerate intersection form.
9.2 Second structure connections
149
To ameliorate that he (essentially) defined the whole series V(j) and proposed
to study the period map corresponding to V@).
Theorem 9.4 below is a translation and generalization of parts of the results
in [SK9, §5], [Du3, Lecture 3 and Appendix G], [Man2,11§1]. Other properties
of the connections V(j> will be discussed in the following sections.
Definition 9.3 Denote M := С x M - V. Fix s e С The second structure
connection Vw on pr*TM\^ -> M is defined by the following formulas for
X, Y e pr *Tm \gi (here consider X as a vector field on M and У as a section in
the bundle)
= vxy -
+ i +
Theorem 9.4 (a) The connection
satisfies for X, Y e pr*Tu\^
bis)
(W - zTl(x о у),
(W - zT\Y).
(9.10)
(9.11)
is aflat connection on pr*TM\^ and
V«((W - г)У) = {U - z)VxY - X о
v?>((W - z)Y) = (U-
(b) The endomorphism
-H(y)-
(9.12)
(9.13)
(9.14)
is a homomorphism of flat vector bundles
{р
It satisfies
and especially
{рг*ТМ\й, V'5»).
(V<jf)*y = As о Д,+1 о ... о Д,+*_,(У)
(9.15)
(9.16)
D-l
fork > 1 andY е рг~1(Тм)\м- It is an isomorphism if and only if d-, ^~
s ф Ofor alli=.\,...,m (cf. Remark 9.2 (e)).

150 Fmbenius manifolds and second structure connections
(c) The Ofo -bilinear map
g ¦
, У)
(9.17)
is a symmetric and nondegenerate and multiplication invariant pairing on the
bundle pr*TM\fa and satisfies
giA-^X, У) = -g(X, ASY). (9.18)
IfX is 4(~s)-flat and Y is 4{s)-flat then g(X, У) is constant. Hence the connec-
connections V(~v) and Vw are dual and g induces an isomorphism
{pr*TM\lk,
({рг*ТМ\йТ, (V(J))*),
(9.19)
here
is the canonical induced connection.
(d) Considers 6 |Z>o. The bilinear form /(s) on pr*TM\^ which is defined
by
is (—lJs-symmetric and V^-flat and satisfies
/(I+1)(X, Y) = -7«(A,(X
, Y) = $(Д_, о ... о А,_2 о Af_,(Jf), Y), (9.20)
), AS(Y)). (9.21)
It is called the intersection form of Vw. Of course /@> = g.
(e) The restriction of V(i) to {z} x M — V is torsion free for any z 6 C.
7/ге restriction of V@) to {z} x M — Й й f/ге Levi-Civita connection for the
restriction ofg to {z} x M —V.
Proof, (a) First we prove (9.12) and (9.13). The potentiality (9.2) and LieE(o) =
1 • о give for V-flat fields X and Y
о У) -
[X,E]oY-
^ - 1 - Л
== Lie?(X о У) - Lie?(X) oY+(j-^-s)(XoY)
У)
9.2 Second structure connections
151
= X о У + X о Ые?(У) + Xo(j
sJ (У).
= (W - z)Vx(y) - X о ^V - ^ + s
By C^-linearity in X and У one obtains (9.12). Formula (9.13) is a direct
consequence of (9.11). For the flatness of V(s) on рг*ТМ\й, it is enough to
show
and
- z)Z) = 0 for flat X,y,Z (9.22)
= 0 for flat Х,У. (9.23)
Formula (9.22) follows with (9.12) and (9.2) for flat X, У, and (also flat)
(V-i+j)(Z)from
- ^^((W - z)Z).
Formula (9.23) follows from (9.10)-(9.13).
(b) The endomoфhism V + \ + s maps V-flat sections in pr*TM to V-flat
sections (cf. Remark 9.2 (e)). So it is an endomorphism of the V-flat vector
bundle pr*TM with eigenvalues d, - ^ + s,i = 1,..., m on each fibre.
The endomorphism (U - z) is invertible on pr*TM\g,. Therefore As is an
isomorphism if and only if d; - ^ + * f 0 for all i = 1,..., m. (9.16)
follows from (9.15) and the definitions of v?} and As. It remains to show
(9.15).

152 Frobenius manifolds and second structure connections
We obtain with (9.10) and (9.12) for X € pr*TM\^ and for V-flat Y
- z)Y) =
^ + A
= AS(V^\(U - z)Y)
and in the same manner with (9.11) and (9.13)
(9.24)
(9.25)
The formulas (9.24) and (9.25) extend to arbitrary Y € pr*TM\^, formula
(9.15) follows.
(c) The pairing g is symmetric and nondegenerate and multiplication in-
invariant, because g is symmetric and nondegenerate and multiplication invari-
invariant. Formula (9.18) holds because V is an infinitesimal isometry with respect
to g. The rest follows from the next two formulas (9.26) and (9.27). Here
X,Y,Z € pr*TM\g, are chosen such that Z is any vector field on M, X is a
V(~j)-flat section, Y is a V(i)-flat section.
. Y)+g((U - z)-\X), VZF)
- zrl (z о (v - I -
- J - *) (W - гГЧ*), (U - zT\Z о Г) + g(.. Л
= 0, (9.26)
because V is an infinitesimal isometry with respect to g. Analogously
dzg(X,Y)=0. (9.27)
9.2 Second structure connections
153
(d) The endomorphism A_, о ... о As_2 о As-\ maps V(t)-flat sections to
V^-flat sections. Therefore lM(X, Y) is constant for V(s)-flat sections X and
Y, and /<J) is V(s)-flat, V(i)(/(s>) = 0. It is (-lJj-symmetric because of (9.18).
(e) The first statement is obvious from (9.10). The second follows from (c).
?
Remarks 9.5 (a) Consider an isolated hypersurface singularity and the base
space M of a semiuniversal unfolding. We will see in section 11.1 that some
choice (of an opposite filtration to a Hodge filtration and of a generator of a
1 -dimensional space) gives the following: a Frobenius manifold structure on M,
an isomorphism from the cohomology bundle with its flat structure to pr* TM \ ^
with V(~5\ The form /(?> corresponds (up to a scalar) to the intersection form
on the homology bundle. The map A_| о...оД«_2оДа_1 corresponds (up
to the same scalar) to the topological map from the homology bundle to the
cohomology bundle, which comes from the intersection form.
We have m = ц, and dt = I + a\ — a, and D = 2 — (aM — a\)\ here
a\,..., Ыц are the spectral numbers of the singularity (cf. section 10.6. They
satisfy — 1 < a\ < ... < ct^ < и and a,- + aM+i_,- = и — 1.
The map As is an isomorphism if and only if а, Ф | + s for all i. Therefore
A|+t and A_|_1_A: for к е Z>0 are isomorphisms. All the maps A^+k for
к € Z are isomorphisms if and only if all а,- ф Z, that means, if and only if the
intersection form is nondegenerate. In fact, this also follows from the relation
between A_« о ... о А|_2 о A|_i, the form /(i\ and the intersection form.
(b) Consider an isolated hypersurface singularity as in (a) with degener-
degenerate intersection form. Then the homology bundle has global flat sections: the
monodromy group is generated by the Picard-Lefschetz transformations of a
distinguished basis of vanishing cycles (cf. [AGV2] for these notions). The
elements of the radical of the intersection form in each fibre glue to global flat
sections.
The cohomology bundle never has global flat nonzero sections. Such a section
would give a flat subbundle of rank /x — 1 in the homology bundle such that
the quotient bundle (of rank 1) would have trivial monodromy. Because of the
Picard-Lefschetz formulas then this subbundle of rank /x - 1 would contain all
the vanishing cycles in the distinguished basis. But they generate the homology,
a contradiction.
This implies V(i> Щ V(-J> and also V(f> Щ V@> Щ V(-5>, because V(J>
and V(~5> are dual and V@) is selfdual (Theorem 9.4 (c)). Except for K. Saito
[SK9, §5], nobody in singularity theory considered the intermediate connections
V(s) for s € (—5, §) П ^Z. Can one describe their monodromy groups using
only topological data? Especially for V@)?

154
Frobenius manifolds and second structure connections
(c) In the situation of (b), V(f> Щ V@) Щ V(~5> also follows from another
result of K. Saito [SK5]: Up to multiplication by scalars, the (degenerate)
intersection form is the only flat bilinear form in the homology bundle. This
follows essentially from the connectedness of the Coxeter-Dynkin diagram
[Ga] [La]. The connection V@) has the nondegenerate flat bilinear form g =
/@).
(d) One could consider the shift from MtoP'xMas uneconomic because
of the following:
(i) Because of Liee(o) = 0 and Lie?(g) = 0, the Frobenius manifold is con-
constant along e. One could take locally an m — 1 -dimensional slice U transversal to
e and equip С x U with the structure of a Frobenius manifold locally isomorphic
to M. Because of [e, E] = e the Euler field also extends to С х U.
(ii) Suppose M = С x U. One can recover V(j) and g on рг*ТМ\м from the
restrictions to {0} x M — T>. They are canonically extended from {0} x M — T>
with the flow of Эг + е: With respect to this flow on pr*TM one has Lie
derivatives Lie9z+e(U - z) = 0, Liedz+e(g) = 0, and also %%eY = 0 for Y
withLieeK = Lie3!r = 0.
(iii) In singularity theory, usually the GauB-Manin connection is considered
on the base M of a semiuniversal unfolding, and not on a space Л х М for
some disc А С С
But it is more convenient to have the whole Frobenius manifold as {oo} x M
in the base space P1 xMof Vм. One can treat all vector fields in 7д/ in the
same way and does not have to distinguish those invariant under Liec. Also,
the second and first structure connections are closely related, and for the first
structure connections one does not have this possibility to reduce the dimension.
9.3 First structure connections
Also the first structure connections are defined on the lift pr*TM (cf. (9.8))
of the tangent bundle of a Frobenius manifold (M, о, е, E, g). The following
definition and statements are known and can be found in different versions in
[Du3][Man2][Sab4].
Definition 9.6 Denote M = С* х М. Fix s € C. The first structure connection
Vw on рг*ТМ\м -> M is defined by the following formulas for X, Y e
рг*(Тм)\й (again consider X as a vector field on M and У as a section in the
bundle)
(9.28)
(9.29)
9.3 First structure connections
155
Remark 9.7 If one pulls back the connection V(i) with the involution i :
P1 x M -> P1 x M, (z, t) i->- (-z, t), one obtains a connection j*V(i) on
pr*TM\fy -*¦ M which is given by the formulas
= V3:r +
- (v + l- + s) (Y)-Eo Y.
+ s)
(9.30)
(9.31)
Theorem 9.8 (a) The connection V(i) is flat.
(b) The multiplication by z is an isomorphism of flat vector bundles
(c) IfX is V(s)-flat and Y is i*V<-~l-s)-flat then g(X, Y) is constant. Hence
the connections V^ and ("V*^ are dual.
(d) For each z € C* the restrictions to {z} x M of all the connections V(s),
seC* coincide. These restrictions are torsion free.
Proof, (a) For flat XJ,Ze pr*TM one has
V^V^Z = z V%\Y oZ) = z Vx(r oZ) + z2XoYoZ
oZ)+z2YoXoZ =
here the symmetry Vx(Y ° Z) = Vy(X о Z) follows from the potentiality.
ForflatX, Y G pr*TM onehasV^K = 0, V3z(XoY) = 0,andV(r)isflat,so
-XoEoY,
+ z-EoXoY.
= XoY
These two terms are equal because of the following calculation. It uses
LieE(o) = 1 • о and the potentiality (cf. (9.2))
X о Y + VXaYE = X о Y - UeE(X oY) + V?(X о Y)
= -Lie?(X) oY-Xo LieE(Y) + VX(E о Y) + [E, X] о Y
(b) Formula (9.29) shows that z Y is V(s)-flat if Y is V(j+1)-flat.

156
Frobenius manifolds and second structure connections
(c) The endomorphism V is an infinitesimal isometry with respect to g and
g is multiplication invariant.
(d) This is obvious from (9.28). Q
In this paper we will not make use of the first structure connections. Con-
Concerning them we restrict ourselves to the following remarks.
Remarks 9.9 (a) In Dubrovin's papers the first structure connection V(~5J is a
main tool for studying the Frobenius manifolds. Sabbah [Sab2,4. l][Sab4, VII
1.1] approaches and characterizes Frobenius manifolds by the first structure
connection Ф(~). More precisely, he considers j* V(t~> where j : P1 xM->
P1 x M is the map (z, t) \-> (—±, t). Also Manin [Man2] considers the first
structure connection V(t~>.
(b) All the connections V(l) give the same family of flat connections on the
submanifolds [z] x M for z e C*. The Euler field is not necessary for their
definition. One can regard this family of flat connections as the primary datum
and the extensions along 9Z via an Euler field as mere refinements.
But the definition of the second structure connections requires the Euler field
right from the beginning.
(c) Some of the first and second structure connections are related by Fourier-
Laplace transformations. One can obtain precise informations following
Sabbah [Sab4, V 2]. One has to consider the whole vector bundle pr*TM
and global meromorphic sections, in order to make everything algebraic with
respect to the variable z. Then one can check that the inverse Fourier-Laplace
transformation in the precise sense of [Sab4, V 2.10] gives a correspondence
between Г УA~и and V(l) for those s eC such that V+{ + s-kis invertible
for all A: 6 Z>j.
This is not satisfied precisely for s in (J?Li(~Ф + ^ + Z>i). In the case of
a hypersurface singularity with degenerate intersection form (cf. Remarks 9.5)
the correspondence is valid for s = — !|, but not for s = |.
(d) One can write the defining equations for Vw and V(s) in a different way,
emphasizing their similarity. For X, Y € pr~l(TM) С pr*TM one has
zXoY, s (9.32)
(9.33)
(9.34)
(9.35)
Using (9.13), one can check that (9.34) and (9.35) define Vw.
9.4 From the structure connections to metric and multiplication 157
(e) In the case of a hypersurface singularity f(x0,.. ¦, х„) the second structure
connection V(~f' corresponds to the GauB-Manin connection on the cohomol-
ogy bundle. In fact, a main point of the construction of a Frobenius manifold
for a singularity / is to enrich the GauB-Manin connection to a connection
isomorphic to V'"'.
Analogously the first structure connection V^J corresponds to a con-
connection coming from oscillating integrals. One can also construct a Frobenius
manifold for a singularity / via oscillating integrals and the first structure con-
connection V(~5-'>. Sabbah generalized this to the case of certain global functions
j ¦ у ->¦ С with isolated singularities on affine manifolds Y. We discuss this
informally in section 11.4.
9.4 From the structure connections to metric and multiplication
The first and second structure connections of a Frobenius manifold (M, o, e,
E, g) are defined on the restrictions of pr*TM to the submanifolds M and M
of P1 x M. But pr *TM is a canonical extension to P1 x M of their sheaves of
sections.
One can apply the notions from chapter 8. We will see that the pairs
(pr*TM, Vw) and (pr*TM, V(I)) carry most of the structure of the Frobenius
manifold. The contents of Lemmas 9.10, 9.13, 9.14, and 9.15 are summarized
in figure 9.1 and figure 9.2.
The lower half of the first diagram is only proved for a massive Frobenius
manifold.
First we discuss the metric. The following canonical isomorphisms are im-
important in order to shift information from pr*TM to TM. The sheaf prtpr*TM
of fibrewise global sections of pr*TM is a free ©^-module of rank in, because
pr*TM can be seen as a family of trivial bundles on P1. One has canonical
isomorphisms
Tm = T{0}*M = 7(оо)хм = pr*pr*TM = рг„рг-хГм. (9.36)
Second structure connection V(l) onP'xM
log. pole {oo} xM _,,^i^P V7
P1 x {0}
log. pole_
multiplication
Figure 9.1

irr.
P1 X
log
pole
{0}
. pole
{00}
{0}
x M
x M
158 Frobenius manifolds and second structure connections
First structure connection V^ on P1 x M
multiplication
V+\ + s, V
Figure 9.2
Lemma 9.10 [Man2, Ц 2.1.1] (a) The pair (pr*TM, V(s)) has a logarithmic
pole along {со} х М. The residue endomorphism and the residual connection
with respect to ± are defined on T{oo]xM. Under the identification T{oo]xM = TM
the residue endomorphism is V + \ + s, and the residual connection is the flat
Levi-Civita connection V of g on M.
(b) The pair (pr*TM, V(j)) has a logarithmic pole along {0} x M. Under the
identification Тщ^м = TM the residue endomorphism isV + \ + s, and the
residual connection with respect to z is the flat connection V on M.
Proof, (a) One can use Lemma 8.9 (b). One has to write out (9.10) and (9.11)
for a basis of global V-flat sections of pr*TM, using the coordinate z = - and
(b) Similarly without z.
?
Using g one can read off not only the flat connection V, but also the metric
g from the logarithmic poles along {oo} x M of the pairs (pr*TM, Vw).
In fact, Lemma 9.11 (a) shows that one can read it off also from the poles along
V. Lemma 9.11 (a) and (b) are not profound, but they will be very informative
when we come to our version (Theorem 10.13) of K. Saito's higher residue
pairing in the singularity case.
As usual, the residue of a meromorphic 1-form со on P1 at a point q 6 P1 is
resqco = ~ fy a> where у is a small positively oriented loop around q.
Lemma9.11 (a)LetX,Y eTMandX,Y e pupr*TM be the lifts to pr*TM.
For anyt 6 M
g(X, Y)(t) = r«(oo,o g(X, Y)dz
= - J2 res(Mg(X,Y)dz.
(9.37)
9.4 From the structure connections to metric and multiplication 159
(b) Consider an s e {%>o ond suppose that A := A_s о ... о Д5_| is an
isomorphism (compare Theorem 9.4 (b) and (d)). Then I{s) is nondegenerate
and induces another nondegenerate bilinear form /(~4) on pr*TM\^ by /(-i) =
/(s) о (Д~' x Д~'). The form /(~s) is V('s)-flat. It satisfies for any X, Y as in
(a)
g(X, Y) = (-
, Y).
(9.38)
Proof, (a) The first equality follows from the definition of g. The sum of the
residues of a meromorphic 1-form on a compact Riemann surface is 0.
(b) The form I(~s) is V("j)-flat because of Theorem 9.4 (b) and (d). Formula
(9.38) follows with (9.16), (9.20), and the (-l^-symmetry of I(s). D
For the first structure connections there is no analogon of g. Theorem 9.8
(c) leads to another interplay between metric and the first structure connections.
The following version for V^^ is close to [Sab4, VI 2.b]. A quite different
version is in Dubrovin's papers (e.g. [Du4, chapter 3]). We will not subsequently
use it. But again, it will be informative to compare K. Saito's higher residue
pairings in the singularity case with it.
As in Remark, г : P1 xM -> P1 xM denotes the involution (z, t) \-+ (-z, t);
the induced involutions on pr*TM and OPixW and other sheaves on P1 x M
are all denoted by i*.
Lemma 9.12 Fix a point q 6 {0} x M С P1 x M. Let gq be defined by
gq : (pr*TM)q x (pr*TM)q -+ OrxM,q
It is a nondegenerate and OPi xM-sesquilinear pairing, that is
gq{f -X,Y) = f- gq(X, Y) = gq(X, i*f ¦ Y)
for f 6 OpiXM,?- It is i-hermitian, that means,
gq(Y, X) = i*Cgq(X, Y)),
and satisfies for Z e {pr-{TM)q, X, Y e (pr*TM)q
Z gq(X, Y) = g,(v^X, Y) + gq(X, V^Y)
Эг g,(X, Y) = gq (V^X, Y) - gq {X, V^Y)
Proof. Everything follows easily from the definitions.
(9-39)
(9.40)
(9.41)
(9.42)
(9.43)
D

160
Frobenius manifolds and second structure connections
Now we come to the multiplication. The multiplication of a Frobenius man-
manifold (M, о, е, E, g) is encoded in that pole of the first and second structure
connections which does not encode the flat structure. The following is an ex-
explicit description for the first structure connection. A more general discussion
is given in [Sab4, 0 14.c].
Lemma 9.13 The pair (pr*TM, Vw) has a pole of Poincare rank 1 (in the
sense of Definition 8.5) along {oo} x M. Fix coordinates t\,...,tm on M
around a point q, the coordinate z= \ on (P1, oo), and fix a basis of pr*T M
in a neighbourhood of (oo, q) € P1 x M. The connection matrix of V(s) with
respect to this basis takes the form
1=1
df,-
г
(9.44)
where all coefficients are in Cp
Under the identification Тм = Tjoo) x м, the matrix —f2o(O, t) encodes U and
the matrix ?2,@, t) encodes the multiplication by j^.
Proof: All statements follow directly from Definition 9.6.
?
For the second structure connections, the multiplication is encoded in the
geometry of V and the poles of (pr*TM, ^7(i)) along it. This will be made
precise in the next section in the case of a massive Frobenius manifold.
Here we give only a weak general statement: If X, Y e рг*рг*Тм then also
VXF € pr*pr*TM and ^^Imxm = VxF|(oo)xM. Therefore one can recover
from V^F the fibrewise global section VXY and the difference V^Y —VXY
and then with (9.10) the product X о Y, if V + \ + s is invertible.
9.5 Massive Frobenius manifolds
A Frobenius manifold (M, o, e, E, g) is massive if it is generically semisimple.
It is semisimple if locally a basis of vector fields e\,...,em exists with e,- oe, =
5,-ye,-. Then these vectorfields are unique up to renumbering. They are called the
idempotent vector fields. Semisimple Frobenius manifolds have been studied
thoroughly ([Du3], [Man2], [Hi]).
The potentiality, or more precisely the condition (9.1) for an F-manifold
implies [e,-, ey] = 0 (Theorem 2.11). Coordinates щ, ...,um withe,- = ^ are
unique up to shifts and are called canonical coordinates, following Dubrovin.
An Euler field E with LieE(o) = 1 ¦ о takes the form E = X!/=i(Mi + ri)ei
9.5 Massive Frobenius manifolds
161
for some г,- б С (Theorem 2.11). Therefore the eigenvalues of U(= E о ) are
locally canonical coordinates.
Let us fix a massive Frobenius manifold (M, о, е, Е, g). The bifurcation
diagram В С Mis the set of points where some of the eigenvalues of U coincide.
It is empty or a hypersurface and it is e-invariant, because of LieeW = id. It
contains the caustic К, С M, the set of points where M is not semisimple.
A function whose zero set is the caustic will be considered in section 14.1.
Another important hypersurface is the discriminant
V = {t € M | U is not invertible on T,M].
(9.45)
The geometry of the discriminant was discussed in sections 4.1 and 4.3. It is
very rich. It is a free divisor with sheaf DerM(log T>) = E о Тм (Theorem 4.9).
The tangent hyperplanes to T> at smooth points have only a finite number of
limit hyperplanes at the singular points. All tangent hyperplanes and the limit
hyperplanes are transversal to e (Remark 4.2 (i)).
Consider for a moment a Frobenius manifold of the form M=CxM' with
С x {t'} the orbits of e. Then one has (Corollaries 4.5 and 4.6):
(i) The projection V -*¦ M' is a branched covering of degree m.
(ii) The bifurcation diagram is the set of e-orbits through VSing, the caustic K. is
the subset of e-orbits through those points of Vsing where the singularities
are more complicated than the transversal intersection of local smooth
components of V.
(iii) The discriminant determines the multiplication.
For any massive Frobenius manifold, without assuming M — С х М', one
can recover the multiplication from V С P1 x M, because V and P1 x M are
automatically big enough.
Lemma 9.14 (a) The discriminants V and V are related by V П {0} x M =
{0} x V. The canonical projection V ->¦ {oo} x M (or {z} x Mfor any z e C)
is a branched covering of degree m. V is a free divisor with
DerCxM(logP) = (U - z)(pr*TM\CxM)
е). (9.46)
(b) Let U С М — В be a sufficiently small open subset with canonical
coordinates u\,..., um with E = ]Г,- м,е,-. Then
m
ОПР1 x U = [J{(z,u)eCx U \z = Ui}
i=i
consists of m smooth components which do not intersect and which pro-
project isomorphically to {oo} x V. The standard lifts 7\,..., ?„, to P1 x U of

162 Frvbenius manifolds and second structure connections
e\,...,em are uniquely determined by the following conditions:
(i) Their sum is ^e'i = e'(the lift of e),
(ii) Each 2J is tangent to all components {(z, u) & <C x U \ z = и j} of V C\
P1 x U with j ф i.
This determines the e, and the multiplication on M.
Proof, (a) One has to regard the definitions of T> and V and needs Theorem
4.9. Compare also Remark 9.5 (d).
(b) The intersection V П P' x U decomposes as described because U С М —
B. The conditions (i) and (ii) are obviously satisfied, (ii) because of 2) (z — Uj) =
Oforj #i.
On the other hand, 2J is not tangential to {(z, и) е С x U \ z = «;} because
of e'iiz — ui) = — 1. Therefore e"\,.. .,e"m are uniquely determined by (i) and
(ii) and by the fact that they are in рг*рг*Тм. D
Lemma 9.14 (b) characterizes the lifts to pr*TM of e\,... ,em as vector fields
on P1 x M by their relation to V. Remark 9.16 (d) will characterize them as
sections of pr*T M by their relation to V(l). In the case of a massive Frobenius
manifold Lemma 9.10 (a) is supplemented by the following.
Lemma 9.15 [Man2, II 2.1.1] Let (M, о, е, E, g) be a massive Frobenius
manifold.
(a) The pair (рг*Тм, V^') has a logarithmic pole along T>. The residue en-
endomorphism along any smooth piece of V has eigenvalues (—E +s), 0,..., 0).
For s ф — | it is semisimple. For s = — | it is 0 or nilpotent with one 2x2
Jordan block.
(b) The monodromy of V*s^ around a smooth piece of T> is semisimple with
eigenvalues (—e2*", 1,..., I) for s ? \ + Z. For s e | + Z й is the identity
or unipotent with one 2x2 Jordan block. "*
Pwof. Equation (9.46) together with (9.10) and (9.11) show that pr*TM is
invariant with respect to V(s) for any logarithmic vector field. By Definition 8.5
and (8.9), then (pr*TM, V(i)) has a logarithmic pole along V.
For the residue endomorphism we have to be more explicit Let U С M — B,
hi, ..., um, E — J^uiei, ЙПР1 x(/, and ё\, ...,em be as in Lemma 9.14
(b). In a neighbourhood of one component {(z, u) \ z — и,- = 0} of V Л P1 x U
(z - щ)Ъ?% = -(v+\+s) ^^ej. (9.47)
V ^ / z — uj
For j ф i locally this is contained in (z — M,)pr*7^f. Because V is
9.5 Massive Frobenius manifolds
163
skewsymmetric, V(?J) 6 Yljфi
- Therefore the eigenvalues of the
residue endomorphism are (—(| + s), 0,..., 0). The remainder of (a) and (b)
follows from Example 7.11 and section 8.2. ?
Remarks 9.16 (a) In the case of 5 € \ + Z, it is not easy to say when the local
monodromy of V(l) around a smooth piece of V has a 2 x 2 Jordan block and
when it is the identitiy.
In the situation in the proof it has a 2 x 2 Jordan block for 5 = — 5 if and
only if V(e,) ф 0. This is often satisfied. For example, if the discriminant is
irreducible one has essentially only one local monodromy and it has a 2 x 2
Jordan block if and only if V ф 0. But in the case of the trivial Frobenius
manifold A™, that is,
e,- = —, eioej=SiJ, ?=
OU
g{elt
(9.48)
one has V = 0, and the discriminant is {u \ u\ ¦.. .-um = OJ.For.? 6 — \
it is even more difficult to give precise conditions.
(b) For 5 ф 5 + Z the locally free OcxAf-module рг*ТмIcxM is the unique
coherent and reflexive extension of pr*TM\gf\o<Cx M with logarithmic poles
along T> such that all residue endomorphisms along smooth pieces of T> have
eigenvalues (—(j + s), 0,..., 0). This follows from chapter 8.
(c) The situation in the proof has been studied extensively in [Man2, II §3]
for V@). One needs to know only the restriction of (pr*TM, V@), g) to one slice
P1 x (m@)}. Then one can recover the whole structure because (рг*Тм, V<0))
is the unique extension to a flat connection with logarithmic poles along (V П
P1 x U) U {со} x M. The metric g is the V@)-flat extension of its restriction
to P1 x {m@)} - V. The data for the slice P1 x {м@)} are called special initial
conditions. They completely determine the Frobenius manifold locally. In order
to recover the metric g one needs Lemma 9.11 (a).
(d) In the situation in the proof, the coordinate z — ui for the component
((z, u) I z — ui = 0} of V П P1 x U induces spaces C"^ of elementary sections
ofordera for Vwnearthiscomponent,as in(8.13).Herea e ZU— (|+.s)+Z.
The sections ci,..., e~m decompose locally uniquely into a sum (in gen-
general infinite) of elementary sections, their elementary parts. For j ^ i + Z
the fibrewise global sections 7\,... ,e"m satisfy the following and are uniquely
determined by it:
they do not have elementary parts in C^ for a 6 —(| + s) + Z<0 and any
i. The section 2; is the only one of them with a nonvanishing elementary
part in C"^0.
This follows from the residue endomorphism. There is a similar statement for

164
Frobenius manifolds and second structure connections
s € \ + Z, but it is more complicated and less satisfactory because of Remark
9.16 (a).
In all examples of massive Frobenius manifolds which I know, the eigenvalues
of V are rational numbers. A partial explanation is given by the following
application of the second structure connections and of a result of Kashiwara
[Kasl].
Theorem 9.17 Let (M, o, <?, E, g) be a massive Frobenius manifold with a
point t e M such that all eigenvalues ofU : T,M ->¦ T,M coincide. Then the
eigenvalues ofV and thus also the numbers d\,..., dm and D are rational.
Proof. Kashiwara calls a constructible sheaf quasiunipotent if for any map from
a sufficiently small disc to the base manifold the induced monodromy around
0 G disc is quasiunipotent.
He showed that a constructible sheaf is quasiunipotent if its restriction to the
complement of a codimension 2 subset is quasiunipotent [Kasl, Theorem 3.1].
We can apply this to V@) onCxM, taking Vsing as the codimension 2 subset.
The monodromies around smooth pieces of V are quasiunipotent, so all local
monodromies on discs embedded in С х М are quasiunipotent.
Ifz@) denotes the single eigenvalue of U : T,M ->• T,M, thenPnP1 x{t} =
{(z@), 0}- The monodromy on С - {(z@), t)} is quasiunipotent. Its inverse is
the monodromy around {oo} x M with residue endomorphism V (Lemma 9.10
(a)). Its eigenvalues are the numbers dt — у, i = 1,..., m with d\ — 1. D
The assumption for t and U is satisfied for example, if the algebra T,M is
irreducible, that means, local (Lemma 2.1). Then E\t is a sum of a multiple of
e\t and a vector in the maximal ideal of T,M.
Chapter 10
GauB-Manin connections for
hypersurface singularities
Section 10.1 resumes the definition of a semiuniversal unfolding of a singularity
and the resulting structure of an F-manifold on its base space M. This was
treated in greater detail in section 5.1. The cohomology bundle is discussed
from the point of view of chapter 8 in section 10.2. Sections 10.3, 10.4, and
10.6 put together most of the known results on the GauB-Manin connection of
(the semiuniversal unfolding of) a singularity. The presentation is as plain as
possible. The detailed discussion of the extensions H(k) for к > 0 (Lemma 10.2,
Theorem 10.7) is new and leads to a very explicit approach to the microlocal
GauB-Manin system (Theorem 10.10).
Also instructive and not well known are two alternative descriptions for
K. Saito'shigherresidue pairings, both under some restrictions (Theorem 10.13,
Theorem 10.28). The second one gives the link to a polarizing form on the
cohomology of the Milnor fibre. The definition and general facts from [He4]
on polarized mixed Hodge structures are provided in section 10.5.
10.1 Semiuniversal unfoldings and F-manifolds
Let / : (C+1, 0) —*¦ (C, 0) be a holomorphic function germ with an isolated
singularity at 0. Its Milnor number /л е N is the dimension of the Jacobi algebra
0/7/:=Ос.+..о/(|?. ¦¦•!?)¦
Unfoldings of /, morphisms between them, semiuniversal unfoldings, and
the germ of an F-manifold which belongs to / have been discussed in section 5.1.
A semiuniversal unfolding of / is a holomorphic function germ F : (C"+1 x
C^ 0) -» (C, 0) with F\(Cn+l x {0}, 0) = / and coordinates Ц,..., ^ on
(О*, 0) =: (M, 0) such that the reduced Kodaira-Spencer map
l A0.1)
a|0 : TQM -+ O/Jf, ^- » Г^Ч(С"+1 х {0},
dtj \_dtj
is an isomorphism.
165

166
Gaufi-Manin connections for hypersurface singularities
The germ (C, 0) С (C+1 x M, 0) of the critical space is defined by the ideal
Jf = (f?, • ¦ •, §~). It is smooth. It is the normalization of the discriminant
Ф, 0) = <p((C, 0)) С (С х M, 0) where
y:(C"+lxM,0)-*(CxM,0), (x,t)^(F(x,t),t). A0.2)
One can choose representatives of all these germs with good properties in the
following way ([Lo2, 2.D], [AGV2, 10.3.1]). First, s > 0 is chosen such that
/-'@) intersects the boundaries dB1'^ of the balls Bn/X = {x e C"+1 | \x\ <
s'} transversally for all 0 < s' < s. Then S > 0 is chosen such that all the fibres
f~\z) for z € A := B\ С С intersect dB''+l transversally. Finally, we choose
в > 0 and define M := B% such that V does not intersect C A) x M and all
the fibres <p~\z, t) for (г,()еДхМ intersect 3B"+1 x {t} transversally.
Then the space X := F~'(A) n(Be"+1 x M) and the maps F : X -» A and
cp : X —> A x M are good representatives with critical space С С Xofcp and
discriminant ?> = (p(C) С А х М.
The projections ргс,м : С -*¦ M and ?> -*¦ M are finite and flat of degree
fi. The Kodaira-Spencer map
(ргс,м)*Ос, X
A0.3)
where X is any lift of X eTMtoX, is welldefined and an isomorphism of free
C^-modules of rank /x. It induces a multiplication о on TM. Then M becomes
a massive F-manifold (Theorem 5.3). Generic semisimplicity follows from the
smoothness of С and corresponds to the fact that for generic t € M the function
F, : X П (B"+l x {t}) -> A
(Ю.4)
has /x A i -singularities. The caustic К. С М is the hypersurface of parameters t
for which F, has less than [z singularities.
^ The field e = a~'(l|c) is the unit field. One could choose F = t\ +
F{x, t2,..., fM); then one would have e = ^-; but we will not need this in the
following.
The field E := a~'(F|c) is an Euler field of the F-manifold (M, o, e)
(Theorem 5.3). The Kodaira-Spencer map in A0.3) gives canonical isomor-
isomorphisms for all t 6 M, •¦
(T;M,o,?|,)= | (-(-) Jacobi algebra of (F,,je), mult., [F,]). A0.5)
\xeSing F,
The eigenvalues of U : T,M -» T,M, X (->• ? о X, are the critical values of F,
for each t € M. They form on M — 1С locally canonical coordinates и i,... ,um.
Locally on M - /C the fields e, = ^ are defined and satisfy e,- о ej = S,ye,-.
10.2 Cohomology bundle 167
There are two discriminants,
?> = (detWr'@) CM with A0.6)
V = 0>(CnF-'(O))C {0}x M
and
V = ^(C) = (det(W - г«й?))~'@) С А х М. (Ю.7)
The critical space С and the restriction С П F~'@) are smooth. The pro-
projections С -» V and С П F"'@) -»¦ X> are generically one-to-one. Therefore
they are the normalizations of V and V, and VС А х MandPcM are
irreducible hypersurfaces.
The function h := detiU-zid) : АхМ ^ С gives an every where reduced
equation for V, similarly detW for V. Both discriminants V and V are free
divisors (Theorem 4.9),
DerM(logX>) = ?oTM = W(TM), (Ю.8)
Here we use notations similar to those in section 9.2: the tangent bundle TM
is lifted to A x M,
n*TM —> TM
A x M
M,
the canonical lifts of the tensors o,U,e to x*TM ai& denoted by the same
letters. Then A0.9) follows from A0.8) and the definition of V, (cf. Lemma
9.14, Remark 9.5 (d)(ii)).
A careful discussion of the isomorphisms between semiuniversal unfoldings
shows the following (Theorem 5.4).
Theorem 10.1 The germ ЦМ, 0), о, е, Е) of the F-manifold and the germs
(D, 0) С (M, 0), Ф, 0) С (A x M, 0) depend only on f, that means, they are
unique up to canonical isomorphism and independent of the choices of s, 8,6
and F.
10.2 Cohomology bundle
Let / : (C"+l, 0) -» (C, 0), F : X -*¦ A, <p : X -» A x M and V С А х М
be as in section 10.1. We make the additional assumption that n > 1. That
excludes only the AM-singularities in one variable (but not their suspensions

-Г . I
168
Gaufi-Manin connections for hypersurface singularities
in several variables). Their GauB-Manin connections and Brieskorn lattices
are exceptional. One can find a treatment of some Fourier duals of their GauB-
Manin connections and also their Frobenius manifolds in [Sab4, Vll 4.b
and 5.c].
In this section the cohomology bundle of <p is discussed using chapter 8 and
an argument of Varchenko. In the next section things will be compared with
the GauB-Manin connection, that is, with sections coming from holomorphic
differential forms.
Good references for the following facts are [Lo2] and [AGV2] (but, of course,
many of the facts are much older, e.g. [Mi]).
Afibre<p-'(z,Oof<p : Д"->. AxM, (x, t) i-»- (F(x, t), t) is singular if and
only if (z, 0 e V. Each regular fibre is homotopy equivalent to a bouquet of /x
n-spheres. The restriction
<p : <p~l(A x M -t>) ->¦ A x M -V
A0.11)
is a locally trivial C°°-bundle. One may call it a generalized Milnor fibration.
The restriction to (Д - {0}) x {0} is a Milnor fibration.
The cohomology bundle
H":= (J
A x M-t>
A0.12)
has rank /x and a canonical flat structure. Its connection is called V.
The cohomology bundle and its monodromy group are essentially indepen-
independent of the choice of e, S, в in section 10.1: first, the fibres of a representa-
representative for some choice of smaller s', 8', в' are deformation retracts of the corre-
corresponding larger fibres of <p by [LeR, Lemma 2.2]; second, the representative
<p : X -*¦ A x M is excellent in the sense of [Lo2, 2.D], and therefore the
monodromy group does not change when one restricts the cohomology bundle
Hn to a smaller base Bxs, x Bg, С Л х М.
The subbundle of H" of the Z-lattices Hn{<p~\z, t), Z) is invariant un-
under the monodromy group. The local monodromy around T>reg is given by
a Picard-Lefschetz transformation: for even n it is semisimple with eigenval-
eigenvalues (—1, 1,..., 1), for odd n it is unipotent with one 2x2 Jordan block; in
both cases the invariant subspace has dimension /x — 1.
Let H be the sheaf of holomorphic sections of H" and i : А х М — V -*¦
A x M the inclusion. With chapter 8 in mind, there are distinguished extensions
of ft to Д x M.
Lemma 10.2 For any к e Z there is a unique extension 7i№> С i*Ti. of Л
to Ax M with the properties: it is an О дхм -coherent and reflexive subsheaf
10.2 Cohomology bundle
169
ofi*Ti., and (Hik\ V) has a logarithmic pole along V whose residue endo-
endomorphism along Vreg is semisimple with eigenvalues (^ — k, 0, ..., 0) for
2yi - к ф 0 and nilpotent with a2x2 Jordan block for ^ - к = 0.
Proof. By Example 7.11 and Theorem 8.7 there is a unique extension of H
to a coherent and, in fact, locally free sheaf on Л х М — Vsing with a log-
logarithmic pole along VKg and residue endomorphism as above. By Theorem
8.11, the direct image under j* for j : AxM — Vsing -» Л х М is the
sheaf П(к). ?
Which of these extensions ft(k) are free Одхм-modules? Following an ar-
argument of Varchenko one can read this off from the restrictions to Л х {0}. Let
(ti, ¦ ¦ •, t,j.) С Одхл* be the ideal which defines Л х {0} and
B(k) := (nik)/(ti,.... fM)ft№))lAX@)- A0.13)
The sheaf B(k) is an extension of the sheaf of holomorphic sections В of H" | Л* х
{0} to Л x {0}. It is a free Од-module of rank /x. Its sections have moderate
growth with respect to the multivalued flat sections of H"\A* x {0}, because
they are restrictions of the sections of 7iw, and those have moderate growth
(cf. section 8.4, [Del]).
We can use the notions of sections 7.1 and 7.2. The germ B^ is a C{z} -lattice
in V'00 and has /x spectral numbers SpiB1^) = (af\ .... aj>), denned by
G.21) and G.23).
The following is essentially due to Varchenko ([Val, ch. 2], [AGV2, ch. 12]).
It is a relative statement and will be complemented by the next section 10.3.
Theorem 10.3 The space H^ is a free O^Mfi-module if and only ifY!t=\
Proof. Consider /x sections ш\,..., «д of H(k) in a neighbourhood of 0 €
Ax M. Choose a basis of Hn(<p~l(z, t), Z) for some (z, t) e Л x M - V
and extend the vectors to flat multivalued sections 5b ..., 5M of the homology
bundle over Л x M — V. Then det2(((a>/, 5,-)),-;) is a univalued holomorphic
function on Л x M - V, because det2 A = 1 for any monodromy transforma-
transformation matrix A e GL(/x, Z).
The function h — d&tQA — zid) gives an everywhere reduced equation for V.
The eigenvalues of the residue endomorphism of Giw, V) along Vreg imply
that
det2(««,,
_ „ . tn-\-2k
A0.14)

T1
•г 1
¦' 'S
170
Gaufi-Manin connections for hypersurface singularities
for some function g e OAxM_f,tlitg0 = OAxM,0- The following statements are
equivalent:
(i) The (jl sections co\,..., co^ form a 6>AxM>0-basis of H^ (especially, then
Ho is a free Сдхд^о-module).
(ii) They form a basis of Ti(k) in (a neighbourhood of 0 in) Д x M - Vsmg.
(iii) The function g does not vanish in (a neighbourhood of 0 in) Л x M—Vsing.
(iv) It has value g@) ф 0.
On the other hand, the definition of the spectral numbers af},..., а?> in
G.21) and G.23) shows that
for some function g € 0AiO and that there exist sections co\,..., <ид with
Finally, h\A = (-z)" because W is nilpotent on T0M. Therefore there exist
sections cot,..., <ыд with g@) ^ 0 if and only if 2 ? af} = /x(n - 1 - 2?).
D
10.3 GauB-Manin connection
Let /, F, <p: X ^ Ax M, V, and tf" be as in sections 10.1 and 10.2 with
n > 1. Holomorphic differential forms yield holomorphic sections in H". The
investigation of these sections and their relation to the flat connection on H" is
summarized in the notion of the GauB-Manin connection for (p.
The GauB-Manin connection for / was introduced by Brieskorn [Bri2]. It
has been generalized to complete intersections (as <p) by Greuel [Gre] and
K. Saito [SK2]. The GauB-Manin connection for singularities has been studied
and applied further by many people. [Mall], [Phi], [Val], [SK6][SK9], [Lo2],
[Od2], [SM3], [AGV2], [Hel], and [Ku] are some references with the character
of a partial survey.
The languages are very different, from an explicit use of integrals to a so-
sophisticated use of ©-modules and ?"-modules. Many of the results which we
have to cite have been proved several times and in different styles. We will try
to present them in the most explicit way and point to more technical ways in
remarks.
Remark 10.4 Usually the GauB-Manin connection for the semiuniversal un-
unfolding of / is developed for the fibration F~'@) -»• M. One can choose
F = h + Fi(x, t2,..., t,,,). Then t\ takes the role of z.
10.3 Gaufi-Manin connection
171
In fact, the cohomology bundle Hn on Л х М — V with its flat structure is
the trivial extension along dz + e of its restriction to {0} x M — V.
We take the uneconomic version with base Л х М because then one can
treat all coordinates in M equally without an uncanonical choice of f j. Also
it fits better to the second structure connections of Frobenius manifolds (cf.
Remark 9.5 (d)).
Let r\ € Щ, 0 and со e fi'j^}- ВУ diminishing е,8,в from section 10.1, one
can arrange that r\ and со are defined on X. The restriction of ц to a smooth
fibre <p~l (z, t) of (p: X -*¦ A x M is closed and can be integrated over cycles in
the homology. One obtains a holomorphic section in H". The germ in 0'*7iH
is independent of any choice of e, 8, в. So there is a welldefined map
fi*,o -»• (J'*^)o- A0.15)
The form со induces a relative и-form, the Gelfand-Leray form, which gives
on each smooth fibre cp~\z, t) a holomorphic и-form $-\<p~l(z, t). It is the
Poincare residue of the form j^j|B"+1 x {t}. Again one obtains a holomorphic
section in H". Again this gives a welldefined map
n+l
A0.16)
This also all works for the restriction to A x {0}, that is, for the Milnor fibration
of / instead of (p. The sheaf of holomorphic sections of the restriction of H"
to Д* x {0} is called В as in section 10.2. Let i0 : A* -> Л be the inclusion.
Then analogously to A0.15) and A0.16) one has welldeflned maps
C"
c+'.o
(io*B)o,
¦ О'о*Д)о>
A0.17)
A0.18)
and the images are called Щ and Hq. They are the restrictions of the images
of A0.15) and A0.16) to (Л х {0}, 0).
An elementary and classical, but crucial task is the analysis of A0.17) and
A0.18) for the case of the A,-singularity / = x$ + ... + x2n. Let <S(z), z 6 A*
denote a A or 2)-valued family of representatives of the vanishing cycles in the
fibres /-'(z). Then one finds (e.g. [AGV2, p. 294])
Г dx0... dxn
JS(z)
Г
Л(г)
xod*! ...dxn = cy ¦ z~
d/
-CfZ 2
A0.19)
/"'(г)
for some с i, c2 6 C*. Therefore the images in A0.17) and A0.18) are V!
V "r where V is defined as in section 7.1.
and

¦f J
172
Gaufi-Manin connections for hypersurface singularities
If / is any singularity with F and <p as above then the fibres cp x(z, t) for
(z, t) 6 f)reg have only A \ -singularities. The discussion of the A i -singularity
and the definition of 7i№) in Lemma 10.2 imply that the images of the maps
A0.15) and A0.16) are contained in 7i^l) and Tif* and that H'Q С В?~° and
Щ С 0Q *• We w'^ not Prove the following theorem, but comment upon the
difficult and simple parts of it and several ways to prove it.
Theorem 10.5 (i) There are equalities H? = B^andH^ = в?~°. The spectral
numbers (af\ ..., a^0)) =: (au ¦ ¦ ¦,a^) =: Sp(f) of Щ are contained in
(— 1, n) П Q and satisfy a,- + aM+i_; = n — I, the spectral numbers of #q are
(ii) The spaces H^ and H{q1) are the images of the maps A0.15) and A0.16).
They are free OAxM,o-modules. The sheaves H@) and H(~n are free OAxM-
modules.
(in) The inclusion Ti(^n С Ti@) is represented by [r]] i-* [dF л tj] for
П e Q"Xi0.
(iv) The covariant derivatives of sections in 7i(-1) andU^ can be described
in terms ofn-forms r\ and n + 1 -forms a> by
Vdz[dFAt1]=[dr1],
[dF 1
—d/? ,
[dF 1
Va/3liM = [Lied/atia] - V8 I — со .
A0.20)
A0.21)
A0.22)
(v)TheoperatorVszyieldsisomorphismsVgz : H'Q -*¦ Hq andVgz : Ho —>¦
nf\ The space Щ is a free C{{d-l}}-module.
Remarks 10.6 (a) The C{z}-lattices Hq and H'a were considered first by
Brieskorn [Bri2]. The lattice H^ is called the Brieskorn lattice. Its spectral
numbers (a.\,..., a^) form the spectrum Sp(f) of the singularity / [AGV2].
(b) On the one hand, Hq is defined via n + 1-forms. On the other hand,
it is determined by the discriminant V С А х М and the monodromy group
of the cohomology bundle, because of Щ = B^ and the definition of B>0
and 7i@). The discriminant determines the singularity up to right equivalence
(cf. Corollary 4.6, Remarks 5.5 (iv) and (v), [Sche2], and [Wir]). One may
ask how much information H^ contains. This leads to Torelli type questions
(cf. [Hel]-[He4] and section 12.2).
(c) Parts (iii) and (iv) are not deep. Part (iii) follows from the definition of
the Gelfand-Leray form, (iv) can be proved with the residue theorem of Leray
(cf. for example [Bri2]).
10.3 Gaufi-Manin connection
173
(d) Formula A0.20) shows that V3; : H^ -* Щ and V3: : ft*,"" -» ?^0) are
surjective. The injectivity follows from the algebraic descriptions in A0.23)-
A0.26) below.
The injectivity of V3; : Щ -»¦ Щ also follows from H^ с V>0, which is
part of (i) and which was proved by Malgrange [Mall]. Together with Lemma
7.4 this shows that Щ is а С{{Эг"' }}-lattice.
The injectivity of Va; : 7i^[) -*¦ H^ also follows from the fact that H'l\Ax{l]
for t generic does not have global flat sections, see Remark 9.5 (b). We will
come back to it in Theorem 10.7.
(e) The relation (a',',..., a?~") = Sp(f) + 1 in (i) follows simply from
the isomorphism Vs, : H$ -*¦ Hq. But the symmetry a, + aM+i-; = и — 1 and
also — 1 < ot\ < ... < <Хц < п are profound. There are two ways to prove
them. One way uses Varchenko's mixed Hodge structure, which comes from
Hq, see Remark 10.31. The other way uses K. Saito's higher residue pairings,
see Theorem 10.28 (v).
(f) Part (i) gives the identity ][]«; = д ^-. In view of Theorem 10.3, it
implies (ii) and is equivalent to the freeness of Тц* as an 0дхм,о-пкх1и1е.
Varchenko [ Val] [AG V2] gave a proof of the freeness of H^ and thus also of
this identity, which is simpler than either of the two ways to prove (i).
First, one has to somehow see it for the special singularities x$ H h jc^.
Then he shows that any singularity turns up in a semiuniversal unfolding of
such a special singularity. Using determinants as in A0.14) he proves that the
freeness of Ti^ for x$ + ¦ ¦ • + x^ implies the freeness of Ho' for the other
singularity.
(g) Greuel [Gre] gave the first proof that the images of the maps A0.15) and
A0.16) are free 0A)<M,o-modules. In view of the discussion before Theorem
10.5 this implies that the images are 7ц~]) and Ti0 \ He also determined the
kernels (see also [SK6] [SK9] [SM3]).
We set S := A x M, S' := A x M - V, X' := (p~l(Sr). Then
A0.23)
A0.24)
A
""'
л
A0.25)
A0.26)
The morphism <p : X —v A x M is Stein. This allows cohomology classes to be
represented in smooth fibres^)"'(г, f)by fibrewiseglobalholomorphicn-forms.

174 GauS-Manin connections for hypersurface singularities
Greuel and K. Saito obtained
H@) =S
A0.27)
00.28)
Coherence and freeness of these extensions of the relative de Rham cohomology
for <p : X' —> S' are profound results. But we will not explicitly make use of
the precise form of the denominators in A0.23)-A0.26).
The next result, Theorem 10.7, shows which of the (Од х м -coherent) sheaves
H{k) from Lemma 10.2 are free OA xM -modules and extends part (v) of The-
Theorem 10.5 to all of the sheaves H{k). For к < 0 this can be found essen-
essentially in [SK6][SK9], but not for к > 0. We give a proof. But for the fact in
part (c), that H^ is a free Ом,о{{д~1}}-той\А& of rank ц,, we have to refer
to [SM3].
/=o
Лч
i=0
A0.29)
is the ring of microdifferential operators of order < 0 at (dz, 0) б Т*( Д x M)
[Phl][SM3].
As in A0.10), 7Г : Д x M -+ M is the projection. The sheaves nJiP of
fibrewise global sections are OM-modules, but not coherent 0w-modules, they
are too big.
Theorem 10.7 (a) The covariant derivative Va.. maps W(t) surjectively to
for any tgZ. For any к е Z it yields an isomorphism
of Ом-modules and an isomorphism
A0.30)
A0.31)
ffej The sheaf V.^ is a free OAxM-module if and only if it is a locally free
OAxM-module and if and only if the germ H^ is a free OAxMfi module.
If a, <? Zfor all i then all H^ are free О АхМ ^-modules. Else, K^ is a free
OAxM,0-module if and only if к < min(a,- | a, 6 Z).
(c) Each germ H^ is a free О Mt0{{d~1}}-module of rank ц.
10.3 Gaufi-Manin connection
175
Proof, (a) From the definition of H(k) it follows that V3; maps H{k) |(Д х М -
Vsmg) surjectively to H(k+l)\(& x M - Vsing). Therefore V3; maps H(k) to
H<~k+I), but the surjectivity near Vsing still has to be proved.
For A0.30) and A0.31), first consider H"\(A x {f} - V) for some generic t.
Then Ft has ц. А \ -singularities and ц. critical values. We claim:
(i) Any global holomorphic section a in Hn |(Д x {/} — V) is the image under
Va. of a unique holomorphic section in H" |(Д x {f} - V).
This claim shows that A0.30) and A0.31) are injective. It extends Remark 9.5
(b) that Я'!|(Д x {t) - t>) has no nonzero global flat sections and follows in
the same way.
Proof of the claim: One chooses a system of nonintersecting paths (e.g. also
ordered anticlockwise) in Д x {r} - V from a regular value to the ц, points
in Д x {f} П V. One obtains a (distinguished) basis of vanishing cycles. The
monodromy around a point (z, t) e Д x {f} П V is given by a Picard-Lefschetz
transformation.
Therefore the space of preimages of a under Va2 in a neighbourhood of (г, t)
in Д x {f} П V forms ад-l dimensional affine space. The difference of two
preimages is contained in the д — 1 dimensional vector space of flat sections
which vanish on the vanishing cycle that corresponds to (z, t). One extends the
д affine spaces of dimension /л — 1 of local preimages of a under Va; along
the paths to the regular value. There they intersect in one point because the
vanishing cycles form a basis. This shows the claim. 0
Now suppose that a e 7Г*?^№+1), that is, a is a section of Tiik+1) defined
in ti~x{U) for some open subset U С М. Let Вм С М be the bifurcation
diagram, the set of t e M such that F, has less than /x different critical values.
It contains the caustic 1С, the set of parameters t e M such that F, has less than
д critical points. We claim:
(ii) The unique preimages under V3; in generic slices Д x {r} - V as above
glue to a holomorphic section in n~[(U — Вм) — i>.
(iii) The holomorphic section is in я»^*\
(iv) It extends to n~\M - /С),
(v) It extends to ж ~' (M).
(ii)-(iv) are easy to see. But unfortunately, (v) is not so clear a priori. We claim
that it follows from

176
Gaufi-Manin connections for hypersurface singularities
(vi) The map in A0.31) is an isomorphism in the case of an ^-singularity
Proof of (vi) =>¦ (v): It is sufficient to show that the preimage under V3. of
the above section a extends to slices Д x {t} — V for generic t e. 1С. Then it
is defined outside of a codimension 2 subset of n~l(U) and extends globally.
For generic r@) e K, the function F,v» has /x — 2 A \ -singularities and 1 A^-
singularity and /x — 1 critical values. One can extend the proof of claim (i) if
one can show that a has a preimage under V3_ in a neighbourhood in Д x M
of the critical value of the ^-singularity.
In such a neighbourhood one can split the cohomology bundle into a subbun-
subbundle of rank 2 for the versal unfolding of the A2-singularity and into a subbundle
of rank /x — 2 which is invariant with respect to the local monodromies. The
subbundle of rank 2 with its fiat connection is induced by the cohomology bun-
bundle of a semiuniversal unfolding of the A2-singularity. Therefore (vi) would
give locally a preimage of a under Va, and would allow the proof of claim (i)
to be applied. 0
The eigenvalues of the monodromy of an ^-singularity are ±i for any
number of variables. Therefore (vi) follows from the claim
Ak)
Ak+\)
(vii) The map Vg2 : Tl0 —> Tl0 is an isomorphism in the case of a singu-
singularity with a,- $ Z for all i.
Proof of claim (vii): When к < —1 it follows from Theorem 10.5 (v) and
from claim (iii) (in fact, without any condition on the spectral numbers).
When к > 0 one can argue as follows. Let a\,..., aM be an Одхл^о-basis
of Tif\ Their restrictions to (Д x @},0) generate the C{z}-lattice Щ' with
spectral numbers a\,..., aM. The restrictions to (Д x {0}, 0) of the derivatives
Vg+1cr|, ..., Vg+1CTM generate a C{z}-lattice with spectral numbers at\ —к —I,
..., ад — к — 1. Because of claim (iii) and Theorem 10.3 this C{z}-lattice is
4*+1),and?4f+1) is a free 0AxM,o-modulegeneratedby V^+'ct,, ..., v?+l<rM.
By induction on k, starting with к = — 1, one can show that V^H^ is an
Одхм.о-module. Then V*+1?C = VbiHf = W<f+1). ' 0
(b) The first statement is clear because Д x M is Stein. The second statement
has been shown in the proof of claim (vii).
Suppose that there exist a,- e Z. Because of (a) we have B(t) = V* Щ for any
к € Z (with Bw С V>0 for к < -1). Looking at the derivatives of principal
10.3 Gaufi-Manin connection 177
parts, one sees easily that the spectral numbers a\\ ..., a^ of Bik) satisfy
(af\ ...,a™) = Sp(f) -k fork< min(a,- | a,- € Z), A0.32)
^p - *)
One can apply Theorem 10.3.
min(a, | a, € Z). A0.33)
0
(c) In the notation of [SM3, §2], Hf is the germ (F_n J° OxH of the filtered
GauB-Manin system for cp : X -*¦ A x M which is constructed there. By
[SM3, §2 E.17) and E.18)] it is a free 0M,o((t' }}-module of rank ц. Because
of the isomorphism A0.31), then all germs Hq are free 0м,о{{Эг Х }}-modules
of rank /x. This finishes the proof of Theorem 10.7.
?
The following considerations on the quotients W(t)/W(* l) are due to K. Saito
[SK6][SK9] (for к < 0). Because of Theorem 10.5 (iii), A0.27), and A0.28)
there is an exact sequence
0 —> H{~1)
Here r@) is the projection.
0.
A0.34)
tt л
is the sheaf of relative (n + l)-forms with respect to <p : X -> Д x M. Its
support is the critical space С С X of <p. It is a free Oc -module of rank 1. The
sheaf
QF :=
A0.35)
is a free 0M-module of rank /x. Via the Kodaira-Spencer isomorphism a in
A0.3) it is a free 7^-module of rank 1. We write the action as
(X, [w]) i-> a(X) ¦ [a] for X e Тм, [со] е fif. A0.36)
Vector fields X e TM will be identified with their canonical lifts to Д x M,
that is, with the fibrewise global lifts X which satisfy X(z) = 0.
Corollary 10.8 For any к € Z one has the following,
(i)
V :
(-k+i'>
A0.37)

178 Gaufi-Manin connections for hypersurface singularities
(ii) There are exact sequences
where r(i+" := r<0) о Vr*
O-
(Hi) IfX € TM and a e пшН{к) then
^(VxVg-'a) = -a(X) ¦ rlk\a).
A0.38)
A0.39)
(iv) Suppose that v{ e H^ is represented by a volume form u(x, t)dx0 ...dxn
€ Q"^ with м@) ф 0. Then the period map
A0.40)
is injective and yields a splitting of the sequence A0.38) for к = -1
Proo/: (ii) follows from Theorem 10.7 (a) and A0.34), (i) and (Hi) follow from
the formulas A0.20) and A0.21) for the covariant derivatives of sections in
H(~[). (iv) follows from A0.39) and the fact that a volume form represents a
generator of f2fi0 as 7^i0-module. ?
Remarks 10.9 (a) By definition of H(k\ the pair {Hik\ V) has a logarithmic
pole along V, thus besides A0.37) one has
V :
A0.42)
One can also see this algebraically [SK6][SK9], using A0.39) and the fact that
precisely the vector fields in DerAxW(logP) are liftable along cp to X (cf. for
example [Lo2, F.14)]).
(b) A primitive form [SK6][SK9] is a section u, in H^ (or in jr*ft@)) which
is represented by a volume form and induces a splitting A0.41) with good
properties with respect to K. Saito's higher residue pairings (cf. sections 10.4
and 11.1).
Our quite elementary approach to the sheaves H(k) and their properties
should be compared with the constructions in [Phi] [Od2] [SM3]. The fol-
following theorem comprises the relations. A detailed proof would require precise
definitions of all the notions and objects, from which we refrain. We give
only a rough sketch. We refer to [Phi] [KK] [Bj] and to the survey articles
10.4 Higher residue pairings
179
[Odl] [No] [Kas2] for the definition of regular holonomic ©-modules, char-
characteristic varieties, and good filtrations. We will not subsequently make use of
Theorem 10.10.
Theorem 10.10 The union {JksZ^k) 's a reSular holonomic VAxM-module
with characteristic variety ГдхМ(А x M) U T?(A x M) and good filtration
Ti^- Asa ?>дxм-module it is isomorphic to the microlocal Gaufi-Manin system
constructed in [Phi] [Od2] [SM3].
Sketch of a proof. The ©-module \Jk€Z H(k) is a local system on A x M - V
and singular along V. Therefore its characteristic variety contains the zero
section T*xM(A x M) С Г*(А x M) and the conormal bundle 7^(A x M)
of V, that is, the closure of T? (A x M).
In order to see that this is all, first one has to check that 7i(*' is a good
filtration. This follows from Theorem 10.7 (a). Then one has to study the ideal in
Ot'm which is defined by that filtration and which determines the characteristic
variety. It has homogeneous (in the fibre variables) generators of degree 2 and
degree 1.
The generators of degree 2 come from the defining relations ^- о ^- =
J2k au э7 f°r tne multiplication in TM and from A0.39). The zero set of the
ideal of these generators is the zero section and the д + 2 dimensional union
of the conormal bundles of all shifts of V along Эг. The generators of degree
one come from the logarithmic vector fields. They cut out the zero section and
Ц(А x M) (see also [SK9, p. 1254], [Od2, B.3)]).
Therefore {JkeZ H(k) is a holonomic Рдхм-module with characteristic va-
variety as claimed. It is regular holonomic because the subsheaves Ti1-^ have a
logarithmic pole along V.
The microlocal GauB-Manin system [Phl][Od2][SM3] is the unique exten-
extension of Hm to a VAxM module (J*>o 9*^@) such Ла1 9г is avertible on the
germ at 0. Because of Theorem 10.7 (a), the union (J*>o ^(t) *s isomorphic to
?
10.4 Higher residue pairings
Let /, F, <p : X -+ A x M, Я", V, and H(k) be as in sections 10.1 and 10.2
with n > 1. In this section we formulate the properties of K. Saito's higher
residue pairings on я*7^@). For the algebraic construction and proof we refer
to [SK7], for sketches to [SK6] [Nam] [SK9]. Constructions in the framework
of microlocal differential systems can be found in [Od2] [SM3]. There it is

180
Gaufl—Manin connections for hypersurface singularities
emphasized that Kashiwara observed that the higher residue pairings can be
identified with a microlocal Poincare duality.
We will offer two completely different descriptions of the higher residue
pairings. But both are valid only under some restrictions. The first in Theorem
10.13 holds only for singularities with nondegenerate intersection form, the
second in section 10.6, Theorem 10.28, holds only for the restriction to (A x
@), 0). Nevertheless both are instructive and explain the properties of the higher
residue pairings.
For the family of functions Ft, t ? M, and the fixed coordinates xq, ... ,xn
on the fibres of the projection ргм '¦ X —*¦ M, (x, t) н-> t, the Grothendieck
residue associates to a relative n + 1-form со ? (ргм)*?1'^м a function
. Элго'
dF
Bx.
A0.43)
It can be defined algebraically or analytically [GrH]. The value at t ? M
is 1/B7T(O"+I times the integral of the form <w/ff • • • • • fj- over the cycle
{(*, 0 I iff I = К} С B?+l x {t} for a sufficiently "small у > 0. It induces a
welldefined pairing
Jf:nfxfif-> OM,
... dxn], [g2d*0... dxn]) н» Res дум
3F_
1?
Ъхп
A0.44)
= (Ргм)*^х/Ахм (cf- (Ю.35)). К. Saito observed that JF is independent
of the coordinates xq, ..., xn. By Grothendieck it is nondegenerate.
The sheaf QF is not only a free Ом-module of rank fi, but also a free
(Prc,M)*Cc-rn°dule of rank 1. By definition, JF satisfies
for g € (prc,A/)*Oc, [a>i], [«2] ? ?2F.
So, JF is a canonical nondegenerate pairing on QF = 7Г»'И(О)/7Г*'И(~1*
(cf. A0.38)). K. Saito extended it to a series of pairings on ntTiP\ his higher
residue pairings Кр~к\
Theorem 10.11 ([SK7][SK6][SK9]) There exists an Ом -bilinear pairing
KF : n*ni0) x л,П@) -> OM[[Эг-1]] • Э"" A0.46)
KF((O{, @2) = / Кр (со\, (О2) ¦ Эг "
with Kp~к\ш\, ау?) ? Ом and the following properties.
10.4 Higher residue pairings
(i) The pairing Kp~k) is (—1)* -symmetric,
(ii) For w\, саг б
181
(Hi) Fora,b < 0, a + b < -k
and for u>i,a>2 ? тг
(iv) Forш\,шг&
KF(z ¦ co\,a>2) — KF(co\, z ¦ un) = [z, KF(co\, 002)],
where [z, Эг~*] := к Э"*-1.
(v) For ш\,со2 ? nJ-(S~x), X ? TM (X ? TM is lifted to A x M such that
X(z) = 0)
X KF{co\, сог) = KF{Vxa)\, «2) + KF(tou VX(O2).
Remarks 10.12 (a) In view of Theorem 10.7 (a) and the properties of KF, one
can extend KF to a pairing on Ujkez^*^'*' ^ш values in ОдЛСЭ^'ШЭг] and
similar properties.
(b) This extension is studied in [SM3, 2.7]. From the microlocal properties
of the GauB-Manin system one obtains
KF(m,CO2) ? OMfi{\KX)) ¦ К"'' for«,,u>2 ? nf * (n*H@))Q
(c) It is also proved in [SM3, 2.7] that KF is uniquely determined by its
properties. This follows essentially from the fact that the germ (/J Ox)o of
the GauB-Manin system is simple holonomic as a microdifferential system, so
that the only endomorphisms of it are multiplications by scalars ([SM3, 2.7],
[Kas2]).
The intersection form / on Hn(<p~\z, t), Z) for a regular fibre (p~l(z, t)
induces a map Я„(<р~'(г, t), Z) -> Hn(cp-l(z, t), Z), S ^ 1(8,.) [or put the
other way, the intersection form is induced by the canonical map
Hn(<p~\z, t\ Z) ^ Hn(<p~\z, t), d<p~l(z, t), Z) = H\<p-\z, t), Z) ].
The intersection form / is flat on the homology bundle
Я„= (J Hn(<p-\z,t),C)

182
GauB-Manin connections for hypersurface singularities
and induces a homomorphism Hn —> H" of flat bundles. If the intersection form
is nondegenerate then this map is an isomorphism and induces a flat bilinear
form I* on the cohomology bundle H".
The residue pairing JF was defined by residues in the fibres of prM : X -»¦
M. We can present the pairings K^k) by residues in the fibres of n : Д x M ->
M in the case of a nondegenerate intersection form. The following result is infor-
informative, but we will not subsequently use it. The residue at (zo, i) of a meromor-
phic function g,(z) in A x {t} is by definition res{Zo,t)(gt) = ?i f\z-zo\=y 8t(z)<k
for some small у > 0.
Theorem 10.13 Suppose that the intersection form I of the singularity f is
nondegenerate. Let I* be the induced bilinear form on the cohomology bundle
H". Then for w\, «2 e 7r*H@), fixed t e M, and anyk>0
A0.47)
Sketch of two proofs: I can offer two quite indirect proofs, but not a direct proof
which would relate A0.47) to K. Saito's definition of KF.
The first proof uses a result of Varchenko and the result of M. Saito in
Remark 10.12 (c). By [Va6, §3.2, Theorem], the right hand side of A0.47) is
equal to JF{[co\], [«2]) for к = 0. One can and has to check that the series of
bilinear forms defined by the right hand side of A0.47) satisfies also the other
properties in Theorem 10.11. Property (i) holds because I is (— l)"-symmetric.
The properties (ii), (iv), and (v) are simple. The first line of (iii) follows from
the definition of H^ in Lemma 10.2: for generic t (and thus for all t) the
function l*(Vg+kai, co2) is holomorphic on Д x {t} if &>] e Hw, co2 e H{b)
with a + b <~-k (cf. [Va6, §3.2]). Because of Remark 10.12 (c), KF must
now coincide with the series of bilinear forms defined by the right hand side
of A0.47).
The second proof will follow from the construction of a Frobenius mani-
manifold in section 11.1, from the induced isomorphism between (#", V, I*) and
(л*ТМ, V(~ г\ 7<-f' up to a scalar) (cf. chapter 9 for the notations), and from
Lemma 9.11. In order to determine the scalar between 7(~t) and /* one again
needs [Va6, §3.2, Theorem] D
Remarks 10.14 (a) Formula A0.47) explains the properties of the pairing KF,
as one could see in the sketch of the first proof.
(b) One can obtain an analogous formula for the general case of a singularity
with any intersection form if one embeds the Milnor fibration into a fibration
with projective fibres as in [Va6, §3.3]. But then one has to check that things are
10.5 Polarized mixed Hodge structures and opposite filtrations 183
welldefined and independent of choices. We refrain from carrying it out here.
Theorem 10.13 is informative, but we will not subsequently use it in any case.
10.5 Polarized mixed Hodge structures and opposite nitrations
There are several possible definitions of a polarized mixed Hodge structure.
The definition presented in [CaK] [He4] is motivated by Schmid's limit mixed
Hodge structure. It is the correct one in the case of isolated hypersurface sin-
singularities. Steenbrink's mixed Hodge structure [Stn] is polarized in this sense.
One can recover it from the Brieskorn lattice. Varchenko [Va2] first saw that
the Brieskorn lattice induces mixed Hodge structures. In section 10.6 we will
regard this as a feature of the Brieskorn lattice.
Here we want to resume the definition of polarized mixed Hodge structures
and the construction of classifying spaces for them from [He4, ch. 2]. Opposite
filtrations and classifying spaces for them will also be discussed. The choice of
an opposite filtration is necessary for the construction of a Frobenius manifold
in section 11.1.
The weight filtration of a polarized mixed Hodge structure in the sense of
Definition 10.16 comes from a nilpotent endomorphism. Its properties are given
in the following lemma from [Schm, Lemma 6.4], (cf. also [Gri, pp. 255-256]).
Lemma 10.15 Let m e N, Hq a finite dimensional Q-vector space, S a
nondegenerate bilinear form on Hq, S : Hq x Hq -> Q, which is (— l)m-sym-
l)m-symmetric, and N : #q —>¦ Hq a nilpotent endomorphism with Nm+l = 0, which
is an infinitesimal isometry, i.e. S(Na, b) + 5(a, Nb) = Ofor a,b <= Hq.
(a) There exists a unique increasing filtration 0 = W_i С Wo С ... С W2m =
Hq such that N(Wt) С Щ-г and such that Nl : Gr^+/ -> Gr^_, is an
isomorphism.
(b) If I +1' < 2m then S(Wh Wr) = 0.
(c) A nondegenerate (— \)m+l-symmetric bilinear form Si is welldefined on
Gr™+1 for I >0by the requirement: 5/(a, b) = S(a, N% ifa,% e Wm+i
represent a,b e Gr^+(.
(d) The primitive subspace Pm+i(Ho) of Gr%+1 is defined by
Pm+l =ker(Nl+l :Gr^+,-
if I > 0 and Pm+, = 0 if I < 0. Then
= O^NlPn
rm+/
A0.48)
A0.49)
and this decomposition is orthogonal with respect to Si if I > 0.

184
Gaufi—Manin connections for hypersurface singularities
Definition 10.16 [CaK][He4] A. polarized mixed Hodge structure (abbrevi-
(abbreviation: PMHS) of weight m is given by the following data: a lattice #z with
Нъ С Hq с #r с Не = Hz ® C, a bilinear form S on Hq and an endomor-
phismTV of Hq such that от, Hq, S, N, W,, Si, Pm+/ satisfy all properties in
Lemma 10.15, and a decreasing Hodge filtration F' on Hq with the properties:
(i) The induced filtration F'Gr^ gives a pure Hodge structure of weight к on
Grf, i.e. Grf = F"Grf ® F*+|-"Grf,
(ii) The endomorphism TV satisfies N(Fp)cFp-\ i.e. N is a (-1,-1)-
morphism of mixed Hodge structures,
(iii) The Hodge filtration and the pairing satisfy S(FP, Fm+l~p) = 0,
(iv) the pure Hodge structure F' Pm+i of weight m +1 on Pm+i is polarized by
5,, i.e. (a) S/(F"Pm+,,F+'+1-"Pm+,)=0,
and 03) i2p-'n-lSi(u, Й) > Oif и 6 FpPm+i П Fm*-ppmM, и ф 0.
Remarks 10.17 (a) The primitive subspace Pm+i carries a pure Hodge structure,
because it is the kernel of the morphism Nl+* : Gr^+; -*¦ Gr^_,_2 of pure Hodge
structures. The strictness of the (—1, — l)-morphism N also implies
F'Grf = 0 FpNJpk+2j and F'N'Pk+2j = NJFp+* Pk+2J.
(b) In Lemma 10.15 the number m could be replaced by some bigger number,
but in Definition 10.16 the weight m is essential for condition (iii). Also the
assumption that S is (—l)m-symmetric is not important in Lemma 10.15, but
essential for (iv) (fi).
(c) Condition (iii) implies (iv) (a), but in general it is not equivalent to (iv)
(a). One can easily see the following. Under the assumption of all conditions
except for (iii) and (iv), condition (iv) (a) for all p and I is equivalent to
S(FP П Wm+i, Fm+1-p П Wm-i) = 0 for all p and /.
(d) The form S is called a polarizing form.
Deligne [De2, Lemma 1.2.8] defined subspaces lp'q for a mixed Hodge
structure, which give a simultaneous splitting of the Hodge filtration and the
weight filtration. They also behave nicely with respect to a polarizing form
([SM3, Lemma 2.8], [He4, Lemma 2.3]).
Lemma 10.18 For a PMHS as in Definition 10.16 let
Ip-i :=(Fpn Wp+q) П (J4 П Wp+q + ^ У4'' П VV«-/-i) ¦ A0-5°)
10.5 Polarized mixed Hodge structures and opposite filiations 185
Then
q \ Я
Fp = G) I1'",
I™,
A0.51)
A0.52)
A0.53)
р+ч<1
U4~\ (Ю.54)
,r-s) = 0 for(r,s)^(m-p,m-q). A0.55)
For p + q >m let Iq'4 be the primitive subspace oflp'4,
I+1 : Ipq -*¦ lm~q~x•т~р-1). A0.56)
Then
S(N4p'q, N4™) = 0 for (r, s,i + j)
A0.57)
-m). A0.58)
Definition 10.19 An opposite filtration for a PMHS as in Definition 10.16 is
an increasing filtration U, on He with
N(UP) С Up
S(UP, Um-i-p) = 0.
A0.59)
A0.60)
A0.61)
and I/, =
Remarks 10.20 (a) The decomposition A0.59) is equivalent to A0.62) and
also to A0.63), see Definition 7.15,
pFqnU4, A0.62)
i. A0.63)
(Ю.64)
(b) Condition A0.60) and condition (ii) in Definition 10.16 imply
N{FP П Up) С Fp
-y
Therefore N and all powers NJ of N are strict morphisms for I/., that is,
Nj(Up) — ImNj П Up-j. The powers N' are strict morphisms with respect to
F* and W, because of Deligne's Ip-q (Lemma 10.18).

186 Gaufi-Manin connections for hypersurface singularities
(c) Condition A0.61) and condition (iii) in Definition 10.16 imply
S(FpnUp,F4nUq) = 0 for р+дфт. A0.65)
The pairing S : F" П Up x Fm~p П Um^p ->¦ С is nondegenerate because S is
nondegenerate.
(d) If there is a holomorphic family of Hodge nitrations, each giving a PMHS,
then an opposite filtration for one PMHS is an opposite filtration for all PMHSs
nearby. Being an opposite filtration is an open condition. This follows from
codim Fp П Up = codim^P + codim[/p.
(e) There exist opposite filtrations. Lemma 10.18 induces a canonical oppo-
opposite filtration ?/^°> by
/'•«.
A0.66)
Uq- i<P
It will be useful in the discussion of the symmetries of a single singularity in
section 13.2. But it will not be useful in the discussion of д-constant fami-
families of singularities in chapter 12. In general the Hodge filtration varies there
holomorphically. One can check that the opposite filtration t/<0) varies antiholo-
morphically. But we need a constant opposite filtration.
Lemma 10.21 The space of all opposite filtrations as in Definition 10.19 for a
PMHS as in Definition 10.16 is an algebraic manifold isomorphic to CN°" for
some Nopp € N.
Proof. The algebraic group
[g € Aut(#c, S, N, F')\g = id on Gr? for all p\ A0.67)
is unipotent. We will see that it acts transitively on the space of all opposite
filtrations. Then this space is isomorphic to №' for some Nopp € N (cf. for
example [Bore, 11.13]).
One starts with a basis af\ ..., a™ of Hc (here ц. := dim Hc) which fits to
Deligne's Ipq, that means, a^ e I^W) for each / and some p(i), q(i) e Z.
It fits to the decomposition Hc = 0p Fp П Up°\ that is, a\0) € Fpii) П Up^iy
O h
One has
Nar =
and
A0.68)
for some v,7, au € C. If U. is any opposite filtration, then there are canonical
isomorphisms
Fp n
A0.69)
10.5 Polarized mixed Hodge structures and opposite filtrations 187
There exists a unique basis a\,..., aM of He which satisfies
щ 6 Fp(i) П Upd) and A0.70)
We claim that ai,..., aM satisfy precisely the same relations with respect to N
Naj = 2^ VijOj and S(at,aj) = air
j
A0.72)
The relation for N follows from p(j) = p(i) — 1 for v-tj / 0, from
iVa,- - ? v,va; 6 t/p@-, = iV (a,- - a?») - ^ vy (a; - af) € F"«,
j J
A0.73)
and from Fp@ П f/p(l-)-i = {0}. The relation for S follows from A0.61) and
condition (iii) in Definition 10.16.
The automorphism of #c which maps a] to a,- is an element of the group
in A0.67) and maps t/^0) to U,. Therefore the group in A0.67) acts transitively
on the space of all opposite filtrations. D
Often the situation is more complicated. In the singularity case one has the
following setting:
(i) a lattice Hz С HQ С HR С H - Hc = Hz ®z C;
(ii) a quasiunipotent monodromy h = hs ¦ hu on Hi with semisimple part
hs, unipotent part hu, and nilpotent part N = loghu on tfQ; the eigenspaces
H\ = кег(й5 — к : H -*¦ H); a weight filtration W. on Hq whose restrictions
to Hi and Нф\ := ©x#i ^ come from N as in Lemma 10.15 and are centred
at n + 1 and n;
(iii) a nondegenerate bilinear form S on Hq (respectively on Hz with values
in Q) whose restrictions to Hi and Нфх are (-1)"+1- and (-1)"-symmetric; S
is monodromy invariant; then N is an infinitesimal isometry;
(iv) an /^-invariant Hodge filtration Fo* whose restrictions to Hi and H^\
form PMHSs of weight n + 1 and n together with the other data.
We also call this sum of PMHSs on Hi and H^i a PMHS. Deligne's Ip>4 and
the I™ are hs-invariant and decompose into eigenspaces {lp-q)x and (/,f'% of
hs. One should consider only monodromy invariant opposite filtrations. Condi-
Condition A0.61) has to be taken with m = n + 1 on H\ and m — n on H^i. Lemma
10.21 also holds for such opposite filtrations.

188 Gaufi-Manin connections for hypersurface singularities
In [He4] the space
Dpmhs := {filtrations F' on Hc \ dim. FpPlx = dimFopP;iX,
F* is /jj-invariant and induces PMHSsof weight и + 1
and и on Hi and Я#1} A0.74)
is defined. It is a classifying space for Hodge filtrations F* giving PMHSs with
the same dimensions as Fq . It is studied together with the group
Gz := Aut(#z, h, S)
and other spaces and groups.
A0.75)
Theorem 10.22 [He4] In the above setting, the classifying space Dpmhs is a
real homogeneous space and a complex manifold. The group Gz acts properly
discontinuously on it. The moduli space Dpmhs I'Gzfor isomorphism classes
of polarized mixed Hodge structures as above is a normal complex space and
has only quotient singularities.
Remarks 10.23 In [He4] a bigger space DPMhs is also defined. It is the space
of the filtrations F* which are similar to Fo* with respect to all conditions except
those involving complex conjugation. Dpmhs is an algebraic manifold and a
complex homogeneous space. The group Gq — Aut(#c> h, S) acts transitively
on it. It is a bundle over a product Dpnm of projective algebraic manifolds with
fibres isomorphic to C"^" for some number Njjhre e N. The space Dpmhs is
the restriction of this bundle to an open submanifold Dprim С Dprim, which is a
product of classifying spaces for polarized pure Hodge structures.
10.6 Brieskorn lattice
We use the notations from sections 10.1-10.3; the function f(x0,...,, xn) is
an isolated hypersurface singularity with и > 1. Its Brieskorn lattice Щ is
the germ at 0 of a distinguished extension to 0 of the sheaf of holomorphic
sections of the cohomology bundle H" | д. x (o) of a Milnor fibration for /. It was
defined as the set of germs of sections of Gelfand-Leray forms of (и + l)-forms,
Щ = ttgl, 0/d/ л йп"с~1 0 (cf. A0.18) and A0.25)). Its relation to the GauB-
Manin connection fora semiuniversal unfolding and some of its properties were
already formulated in Theorem 10.5.
The Brieskorn lattice is a key to the singularity. It has a rich structure and
a long history, which started with [Bri2]. Here we give a concise report on its
structure, but not on its history. We follow [He4].
10.6 Brieskorn lattice
189
The notions of sections 7.1 and 7.2 (e.g. the spaces Ca of elementary sections
and the V-filtration) will be used freely. The space H°° is the At-dimensional
space of multivalued global flat sections on the cohomology bundle Я"|д.х|0)
over the punctured disc Д* (cf. G.2) and G.3)). Now it contains the lattice
#°° = Z" of multivalued sections of cohomology classes in Я"(/~'(г), Z).
Here f~\z) is a Milnor fibre of the representative / : В^+|П/-'(А)-^ Д of/.
In addition to the monodromy h on H°°, there is a nondegenerate bilinear
form S : H™ x Я|° -» Q [He4]. It is less well known than it deserves. It makes
Steenbrink's mixed Hodge structure [Stn] on H°° into a polarized mixed Hodge
structure. It induces a series of bilinear forms on НЦ which coincide with the
restriction to #o °f K- Saito's higher residue pairings. It is defined in A0.78).
The variation operator Var : Hn(f~l(z),Z) -> Hn(f~\z), Z) for a Milnor
fibre /"'(г) uses the identification Я"(/~'(г), Z) = Я„(/"'(г), 9/-'(г), Z)
and maps a relative cycle у to the absolute cycle h(y) — y, where h is a
geometric monodromy on the Milnor fibre which fixes the boundary of it. Var
is an isomorphism and determines the monodromy h and the intersection form
/ on Яп(/-'(г), Z) (cf. for example [Lo2][AGV2]).
The monodromy has a semisimple part hs and a unipotent part hu with N :—
log hu. It acts on Hn (/-' (z), Z), Я"(/"' (z), Z), and H§°. We use the notations
Я~ = кег(/г, - A.) and H? = ф^, Ял°°, and the same for ЯЛ(/"'(г), С)
andЯn(/-1(z),<C).
The intersection form / on Я„(/~' (г), C)^i is nondegenerate. It induces an
isomorphism to Я"(/"' (z), C)^i and a form /* on it.
An isomorphism v : Яп(/-'(г), Q) -*¦ Я"(/"'(г), Q) is defined by
опЯп(/-1(г),<С)_,ь
(Ю.76)
on
The polarizing form 5 on Я"(/~Чг), Q) is defined by
S(a, b) = (-l)n("-1)/2(a, Var о v(b)).
A0.77)
A0.78)
Lemma 10.24 [He4] The form S is nondegenerate and monodromy invari-
invariant. The restriction to Ял(/"'(г), C)^i is (— 1)" -symmetric and is equal to
(_l)«(«-i)/2/* The restriction to Я"(/"'(г), C)i is (-l)"+l-symmetric.
Remarks 10.25 Unfortunately, in [He4] it was not noticed that the monodromy
there is the inverse of that in [Schm]. Therefore here v and 5 on Я" (/"' (z), C)i

> г' ¦)
>"¦'•'ii
190
Gaufi-Manin connections for hypersurface singularities
differ by a sign from those in [He4]. Also, in A0.83) a sign turns up, and in
Theorem 10.30 (i) one needs —N.
The form S on Hn{f~\z), Q) induces a form S with the same properties on
#0°. This form S can be used to define a C{{9.T1}}-sesquilinear pairing
Kf :
3-1
A0.79)
on the space V»-' = ©_1<ff<0C{z}Ca = ф_|<а<0С{{Эг-'}}Са (cf. section
7.1). One needs the homomorphisms ф-а in G.6) from multivalued flat sections
to elementary sections,
Са,
es(A, a),
A0.80)
in order to go from H°° to V >-1. The pairing Kf and its parts Kf~k) with values
in С are defined by the formulas A0.81)-A0.85), here a, p e (-1,0], a € Ca,
Kf{a'b) =
•9
-l
for a = p = 0,
a, b),
(fk\
Kfigia, g2b) = J^ K(fk\g\a, g2b)
A0.81)
A0.82)
A0.83)
A0.84)
A0.85)
Lemma 10.26 [He4] (i) The pairing K(f k) is (-1)*-symmetric.
(ii) For the restrictions ofKf to the eigenspaces one has Kf : С хС' —>¦ 0
fora+p <?Z,a,p> -1, and A> :CaxC^ С-ЗГ"^ is nondegenerate
fora+peZ,a,p > -1.
finj Г/ге pairing Kf and the multiplication by z are related by Kf{z a, b) —
Kf(a,zb) = [z, Kf{a, b)\ where [z, д~к] = кд~к-\
The pairing Kf shares the properties (i), (iii), and A0.84) with K. Saito's
higher residue pairings (Theorem 10.11). Theorem 10.28 will show that it co-
coincides on Щ with the restriction of KF to Я„.
10.6 Brieskorn lattice
191
Now we come to the properties of Щ (cf. A0.18), Theorem 10.5, A0.25)).
Brieskorn [Bri2] showed that its germs of sections have moderate growth (that
means Щ С V>~°°), that they generate the cohomologies of the Milnor fibres
and that the map V8; : Щ -+ Щ from the sublattice Щ С Щ defined by
n-forms is an isomorphism (cf. A0.17), Theorem 10.5 (v)). Malgrange [Mall]
showed Hq с V>0. Together with Lemma 7.4 this gives the following.
Theorem 10.27 (i) The Brieskorn lattice Щ is a free C[z}-module of rank \x,
equivalents C{z}[z']]H^ = V^00.
(ii) The Brieskorn lattice and Щ satisfy Щ С V>0, H? С V>-\ Щ =
^Щ С Щ.
(Hi) The Brieskorn lattice Щ is a free C{{3~' }}-module of rank ц.
The nondegenerate Grothendieck residue pairing Jp : ?2f x Qf —> Ом
from A0.44) for the semiuniversal unfolding F of f induces a nondegenerate
pairing
Jf :
on the /x-dimensional space
A0.86)
A0.87)
The following theorem from [He4] gives the relation between Hg, Kf, J/, and
KF. Remarks on the proof will be given afterwards.
Theorem 10.28 (i) For к > 0, the pairing Kf~k) on Hg is the restriction of К
Saito's higher residue pairing К^к) on Tif (cf Theorem 10.11) to Hg.
(ii) The pairing and the Brieskorn lattice satisfy Kf~k\Hg, Hg) = 0 for
-n<k< -1, i.e. Kf(Hg, Hg) = CU3-1}} • Э-"-1.
(iii) The pairings Kf* and Jf are related by Kf\coi,aJ) = J/dm], {(&Л)
for a>\, o>i € Hg.
(iv) The Brieskorn lattice Hg is isotropic of maximal size with respect to the
antisymmetric bilinear form K^l\ that means, a section a> € V>-1 satisfies
Kfl\co, Hg) = 0 if and only if со е Hg.
(v) Therefore Hg D Vn~l and dim Hg/ V"~l = |dim V^/V'K and the
spectral numbers ai,..., aM are contained in (— 1, n) and satisfy the symmetry
_,- = n —
Remarks 10.29 (a) Part (i) can be found essentially in [SM2, Appendix],
[SM3, 2.7]. But he is not very explicit about the definition of Kf. Another

192
Gaufi-Manin connections for hypersurface singularities
way to check it is with the results in [Va6, §3.3]. Varchenko embeds the Milnor
fibration (and its extension to the semiuniversal unfolding) into a fibration with
projective fibres. He relates the Grothendieck pairing J/ (and Jf) to the prim-
primitive cohomology bundle of this extended fibration and to the nondegenerate
intersection form on it.
He considers formulas of the same type as A0.47), but for the primitive
cohomology bundle. I checked that these formulas induce a bilinear form on
Ti0 with the same properties as K?, that they induce Kf on V>~1, and that
the properties (ii)-(v) hold.
Now one needs M. Saito's result ([SM3,2.7], see Remark 10.12 (c)) that the
form Kf on 7Yq0) is uniquely determined by its properties. One obtains (i) and
a generalization of Theorem 10.13 (see Remark 10.14 (b)),
(b) The existence of a relative compactification of the Milnor fibration to a fi-
fibration with projective fibres and other good properties is due to [Bri2][Schel].
It is not only essential for the proof of Theorem 10.28, but also for that of
Theorem 10.30.
(c) Part (iv) follows from the facts that // is nondegenerate and that Kf is
CUa-'H-sesquilinear (cf. A0.84)). The form Kf satisfies Kf[)(V>-\
Vй) = 0 and induces a nondegenerate form on the quotient V>~x/V".
This shows the inclusions V С Щ and ai,..., ад e (—1, n) and the di-
dimension formula in (v). A refinement of this argument yields the symmetry of
the spectral numbers.
Varchenko [Va2] [Val] [AGV2] showed that the principal parts of the sections
in Hq induce a holomorphic family of Hodge filtrations on the cohomology
bundle, giving a mixed Hodge structure on each fibre. There is a limit Hodge
filtration FVa on H°°. He used a relative compactification of the Milnor fibration
and results of Griffiths, Schmid, and Scherk. His construction was modified
[SchSt] [SMI] [Ph3] to obtain Steenbrink's [Stn] mixed Hodge structure F* on
Я°°. It is polarized by S because of an exact sequence of Steenbrink, connecting
it with the limit mixed Hodge structure of Schmid for a relative compactification
of the Milnor fibration. This was emphasized in [He4] (for the sign in — N in
part (i) see Remark 10.25).
Theorem 10.30 (i) The subspaces
FPH°° := *"'
G.27)) A0.88)
for a e (—1,0], e~lma — X, define an hs-invariant decreasing filtration on H°°
with 0 = Fn+i С F" С ... С F° = H°°. It is Steenbrink's Hodge filtration.
10.6 Brieskorn lattice
193
Together with S and -N it gives a PMHS of weight n on Hg\ and a PMHS of
weight n + 1 on Hf° (in the sense of Definition 10.16).
(ii) The subspaces
HaH? ¦= ir;xz-n+'Grav+n~PHZ (= Fn-Pfwm G.24)) A0.89)
define an hs-invariant decreasing filtration on H°°. Together with the weight
filtration W. from N this filtration also gives a mixed Hodge structure. The
filtrations Fya and F' coincide on the quotients Gr,w (cf. Lemma 7.4 (b)).
Remark 10.31 In [AGV2][Stn] /j. spectral pairs in Q x Z are defined. The
multiplicity of (a, /) e Q x Z as spectral pair is
d(a, I) := dimGr[;-a]Gr?[a+1]tf;
oo
X
A0.90)
for e~2"'a = X. Because of the PMHS and the strict morphism N, they satisfy
the symmetries (any two of the symmetries determine the third)
d(a, l) = d(n-l- a, 2n - I),
d(a, I) = d{2n -I-I-a, I),
A0.91)
A0.92)
A0.93)
The first entries of the spectral pairs are the spectral numbers. It is unlikely that
one can recover all the symmetries only with Theorem 10.28, the strictness of
the morphism N is probably too profound.
The following result of Varchenko will be used in chapter 12.
Theorem 1032 [Va3] The spectral pairs are constant within a ^-constant
family of singularities.
The next result of M. Saito is essential for the construction of a (good)
primitive form in section 11.1. The д-dimensional space Q/ from A0.87) is a
free module of rank 1 over the Jacobi algebra O/J/. Generators for this module
are represented by volume forms in fi?l"+i 0. that is, forms u(x)dxo ... dxn with
м@) ф 0. The space Я/ inherits a finite decreasing filtration V'Q/ from the In-
Infiltration on Hq. The dimension dimGr^?2/ = d(a) is the multiplicity of a as
spectral number. Parts (ii) and (iii) in Theorem 10.33 are corollaries of part (i).
Theorem 10.33 [SM4, 3.11 Remark]
(i) The maximal ideal m/J/ in the Jacobi algebra maps VaQ.f to V>aQ/.
(ii) «i < a2, that means, d(a{) = 1 and dim Gr^""'1^0?,,,., = 1.

194 GauB-Manin connections for hypersurface singularities
(Hi) A form u(x)dx0 ...dxne Q"^}, 0 represents a section in Щ with principal
part in C"] if and only if со is a volume form, that is, и@) ф О.
In [He4, ch. 5] a classifying space for Brieskorn lattices is studied. Its ele-
elements are subspaces ?0 С V> with the following properties.
(or) The subspace jCq is a free C{z}-module of rank fi.
03) The subspace ?0 is a free C{{3-'}}-module of rank /л.
(у) The decreasing /^-invariant filtration F?J. on H°° from G.24) (cf. A0.88))
is in the classifying space DPMHS for PMHSs from A0.74).
(<S) One has K(f~k\?.o, ?o) = 0 for ~n < к < -1.
Theorem 10.34 [He4] The classifying space
Dbl = {А С V*-1 | ?0 satisfies (a), 03), (y), (S)}
A0.94)
for Brieskorn lattices is a complex manifold and a locally trivial holomorphic
bundle prBL : DBL -*¦ DPMhs with fibres isomorphic to CN"L for some NBL 6
N. The fibres have a natural C*-action with negative weights. The group Gz =
Aut(#?°, h, S) acts properly discontinuously on DPMHS and thus also on DBL.
More details on DBL including a formula for NBL can be found in [He4].
Essentially because of Theorem 10.32, there is a period map from the ^-constant
stratum to the space DBL. Such period maps are studied in [Hel] [He2] [He3]
and in section 12.2.
Chapter 11
Frobenius manifolds for
hypersurface singularities
The construction of Frobenius manifolds for singularities is due to K. Saito
and M. Saito, using results of Malgrange. The version presented in section 11.1
replaces the use of Malgrange's results by the solution of the Riemann-Hilbert-
Birkhoff problem in section 7.4 and by the tools in section 8.2. All the other
ingredients from the GauB-Manin connection are provided in chapter 10.
Section 11.2 establishes series of functions which are close to Dubrovin's
deformed flat coordinates. Some use of them is made in chapter 12. In view of
some results of Dubrovin, Zhang, and Givental one can hope that much more
can be found in these series of functions.
Sabbah generalized most of K. Saito and M. Saito's construction to the case
of tame functions with isolated singularities on affine manifolds [Sab3][Sab2]
[Sab4]. But the details are quite different, there one uses oscillating integrals,
and the results are not as complete as in the local case. We discuss this at some
length in section 11.4.
The case of tame functions is important for the following question within
mirror symmetry: Are certain Frobenius manifolds from quantum cohomology
isomorphic to certain Frobenius manifolds somehow coming from functions
with isolated singularities? This is motivated by the results of Givental. A special
case was looked at by Barannikov. In section 11.3 we make some remarks about
this version of mirror symmetry.
11.1 Construction of Frobenius manifolds
Let / : (C+1, 0) —> (C, 0) be a holomorphic function germ with an isolated
singularity at 0, with Milnor number ц,, and with n > 1. The base space of a
semiuniversal unfolding is a germ (M, 0) = (<СД, 0). It can be equipped with
the structure of a massive Frobenius manifold. The multiplication and the Euler
field are unique, but the flat metric depends on some choice. Its existence
195

196
Frobenius manifolds for hypersurface singularities
is highly nontrivial and follows from the existence of a primitive form of
K. Saito [SK6][SK9], which was proved in the general case by M. Saito [SM3],
building on work of K. Saito and many other people. M. Saito used a result of
Malgrange [Mal3][Mal5] on the extension of special bases in the microlocal
GauB-Manin system, whose proof involved the Fourier-Laplace transforma-
transformation of this system. That made it difficult to use the construction of the metrics
on (M, 0) and to work with these Frobenius manifolds.
Below we give a simpler and more explicit version of the construction, with-
without using Malgrange's result. We will also provide precise information as to
which choices have to be made and what they yield. This makes the construction
sufficiently transparent to be subsequently applied.
To describe the choices, we need the space Я°° of multivalued global flat
sections on the cohomology bundle of a Milnor fibration. It is defined as follows
(cf. G.2) and G.3) and section 10.6). Choose e>0 and S >0 as in section 10.1
such that f:Bne+x Л /-'(A) ^ A for A := B\ С С and 5e"+1 С C"+1 is a
Milnor fibration.
The cohomology bundle Я"|д. := ЦбД. Hn{f~\z), С) has rank ц and a
flat structure. The universal covering e : С —> С*, f \-> e2*'* restricts to a
universal covering e : e~l(A*) -*¦ A*. If pr : е*(Я"|д.) -»• Я"|д. denotes for
a moment the projection, then
H°° = {proA\ Aisa global flat section of e*(Hn\A.)} A1.1)
is the /^.-dimensional space of multivalued global flat sections of the cohomology
bundle. It is independent of the choice of the Milnor fibration. It is equipped with
a lattice Я^° С Я°°, a monodromy A with semisimple part A, and unipotent
part А„, a polarizing form S (see section 10.6), and a polarized mixed Hodge
structure with Steenbrink's Hodge filtration F* (see Theorem 10.30).
The spectral numbers au ..., aM e (—1, n) П Q come from the Brieskorn
lattice Hq, but they also encode the dimensions of the spaces GrpFH?° (see
A0.50)), where Ях°° = ker(A, - X). M. Saito's result (Theorem 10.33) that the
smallest spectral number <*i has multiplicity 1 implies
'a"ff% = 1. (П.2)
The notion of an opposite filtration for F* was defined in Definition 10.19.
Monodromy invariant opposite filtrations U. for F* exist. They have to satisfy
A0.61) with m = n + 1 on Я,00 and with m — n on Я?} :
They form an algebraic manifold isomorphic to C^"» for some Nop
(Lemma 10.21). Such a filtration satisfies
)°° fJOO
dim
= 1.
(П.З)
11.1 Construction of Frobenius manifolds
197
Theorem 11.1 Any choice (t/., y\) of a monodromy invariant opposite fil-
filtration U, for F' on H°° and of a generator y\ of the 1-dimensional space
GrP,_a ]Я°?2».-«, induces the structure of a germ of a Frobenius manifold on
(M, 0). Multiplication o, unit field e, and Euler field E are unique. Different
choices of (I/., y\) Sive different metrics g. If (?/., y\) gives a metric g then
(?/., cyOfor с е С* gives the metric eg.
Let Vg be the Levi-Civita connection of g. The flat endomorphism VgE :
Тм о -* Тм.О* X i-»- V^jE, is semisimple with eigenvalues dt = 1 + a\ — a,-,
i = 1 (i, and D = 2 - (aH - c*i) = 2 + 2a, - (n - 1).
This follows from the results of M. Saito [SM3] and K. Saito [SK6][SK9]
on the GauB-Manin connection and the primitive forms. In this section we give
a simplified proof. It uses ingredients from all previous chapters. A rough idea
of the proof was given in section 6.1.
Proof of Theorem 11.1: The Brieskorn lattice H? (cf. A0.18), Theorem 10.5,
A0.25), section 10.6) induces Steenbrink's Hodge filtration F* = F°ig. on Я00
(Theorem 10.30). Here F^l, is the increasing filtration which is associated to
Щ by G.27).
Let ([/., y\) be as in Theorem 11.1. Define a monodromy invariant increasing
filtration U. on Hc by Up := Up+n on Я,°° and Up := Up+n+\ on Я~. Then
F?s and U. are opposite in Я,00, and F?8 and t/.+i are opposite in Н°?{, in the
sense of Definition 7.15.
The cohomology bundle Я" | д. extends uniquely to a flat bundle over C*. The
Brieskorn lattice Hq gives an extension of its sheaf of sections over 0. The filtra-
filtration U. induces an extension over oo. By Theorem 7.17 this twofold extension
Я" is a free Or -module of rank ц. with a logarithmic pole at oo. The residue en-
endomorphism at oo has eigenvalues -a i,..., aM (Theorem 7.17) and is semisim-
semisimple because of N(UP) С Up-\ and Theorem 7.10 (N = loghu as usual).
Let F : X ->• A be a semiuniversal unfolding of / as in section 10.2 with
discriminant Ъ С A x M and cohomology bundle Я" over Ax M -t>. The
cohomology bundle Я" extends uniquely to a flat bundle over С x M - V.
The twofold extension ~H" embeds into a twofold extension Я@) of the sheaf
of sections of Я": The sheaf Ti{0) from Lemma 10.2 is the unique locally free
(Theorem 10.5)-extension over V with a logarithmic pole along Vreg with
(semisimpleforn > l)residueendomorphismwitheigenvalues(i^!-, 0,..., 0).
By Theorem 8.7 U. or (fljf)oo induce a unique extension over {oo} x M with
a logarithmic pole along {oo} x M.
A priori № is a locally free Or хм-module of rank д. But the restrictions
of its sections to P1 x {0} give the free OPi-module Я". By a classical theorem

1TW
. •¦"!
198
Frobenius manifolds for hypersurface singularities
on families of vector bundles over P1 (e.g. [Sab4,15.b], [MaI4, §4]) then ТЩ
itself is a free OpixM-module if M is small enough. Choosing M arbitrarily
small does no harm. So we will always suppose that it is small enough and often
go from the germ (M, 0) to such a representative M.
As in chapter 9, pr : P1 x M -*¦ M denotes the projection. The sheaf pr^H®)
of fibrewise global sections is a free C^-module of rank /x. The restriction to
{oo} x M yields a canonical isomorphism of O^-modules
Pi
(oo)xAf
A1.4)
The latter sheaf is equipped with the flat residual connection with respect to
the coordinate | (see section 8.2). The global sections whose restrictions to
{oo} x Mare flat with respect to this residual connection form a/U,-dimensional
space. In order to study the features of these special sections we need some
choices and notations.
The opposite filtration U. induces a splitting of the Hodge filtration F' on H°°
with specific properties (cf. A0.59), A0.64), A0.65)). This can be shifted to the
spaces C" of elementary sections in Я" | д.х(о) with the maps т/га : Н^„ -> С"
from G.5). The subspaces
Ga :=
П
^«) С Са
A1.5)
have dimensions d(ce) = multiplicity of a as a spectral number. They satisfy
(Kf is the pairing on V>~1 from section 10.6):
(zV9z -a)Ga = ~Ga С VdzGa+x с
A1.6)
p>0
Kf(Ga,Gli) = 0 fora+/8#n-l,
A1.7)
A1.8)
°Ga+P = ($z-pG°+i> fora>-l, A1.9)
-PGa+p. A1.10)
p<0 p<0
In order to describe the situation along {oo} x M we make the following
observations:
(I) The space H°° is canonically isomorphic to the space of multivalued
flat sections on the restriction to (C — Д) x M of the (extended) cohomology
bundle Hn.
11.1 Construction of Frobenius manifolds
199
(II) On Hn\(C — Д) x M one has elementary sections with respect to the
coordinate z. They are defined by the same formula, compare G.5) and (8.13).
An elementary section a of order a satisfies
(zV3l-a)"+1or =0 and Vxct=0, A1.11)
where X e TM is lifted to P1 x M in the canonical way with X(z) = 0.
(III) The elementary sections over Д * x {0} and those in (II) glue to elementary
sections over C* x {0} U (С-Д)хМ. From now on the spaces С denote spaces
of such global elementary sections. But we will still take the liberty to embed
the С into spaces of germs of sections at (Д x {0}, 0) or (P1 x M, (oo, 0)).
One will usually see from the coefficients what is meant.
By the construction of the extension over {oo} x M, one has
- Д) x M =
A1.12)
Therestrictionto{oo} x M yields a canonical isomorphism of free Cj^-modules
of rank fi
\
a \ I Z /l(oo}xM.
By definition of the residual connection on the right module, the flat sec-
sections of this residual connection correspond to the sections in фа Ga on the
left.
Now we choose a basis s\,..., s^ of фо Ga with
Si e G°",
d-"-1,
A1.14)
A1.15)
Я™*,,,,. A1.16)
Note that dimG = 1 and that s\ is uniquely determined by A1.16). The
sections
v, :=
A1.17)
for i = 1,..., [i form a basis of the space of global sections whose restrictions
to {oo} x M are flat with respect to the residual connection at {oo} x M. Their
germs at (oo, 0) satisfy
A1.18)
here ? := j with "z^j = -zdz.

200
Frobenius manifolds for hypersurface singularities
In the following, if X e TM we will also denote by X the canonical lift to
P1 x M with X(z) - 0; this lift also satisfies [X, dz] = 0.
The operator V^' is welldefined for sections in С with а ф — 1 and, by
Theorem 10.7, for sections in H^ and ntHi0\ where ж : A x M -> M is the
projection. By the proof of Theorem 10.7, V^1 is also welldefined for fibrewise
global sections T^
A1.19)
A1.20)
Lemma 11.2 (a) For X eTM
(b)
z ¦ vt € ф OM ¦ vj + (a,- +
' vt.
Proof, (a) By Corollary 10.8,7Г*7^@) and the sheaf of fibrewise global sections
of ?t@)|cxM are invariant under V^V^.
For the germs at (oo, 0) one has to observe the following. The germ 7^0)(oo,o)
has a logarithmic pole along {00} x M. It contains C°, because its spectral
numbers are —ct\ —aM € (—n, 1). If the monodromy has eigenvalue 1, i.e.
С0 Ф 0, then the germs in z" • W@)(oo,o) do not have unique preimages under
V3j, but any preimage is contained in W^oo.o)- Therefore the germs at @0,0)
satisfy
'-I. A1-21)
A1.22)
Thus Vy V^"' «,¦ is a global section in W®.
(b) One has to apply zVdz - (сц + 1) id to A1.21) and observe A1.6). Then
at @0, 0)
z ¦ Vi - (a,- +
' Vi €
A1.23)
This section is in any case in 7^@)|cxM- Therefore it is a global section in
Wl D
In M. Saito's approach, the equations A1.19) and A1.20) follow directly
from Malgrange's result. Formula A1.19) is used to prove Lemma 11.1 (b) (cf.
[SM3, 4.3]). Again KF is K. Saito's higher residue pairing (Theorem 10.11),
and Kf is the pairing on V>-i defined in section 10.6.
11.1 Construction of Frobenius manifolds 201
Lemma 11.3 (a) The restrictions vf € Щ of the sections vt satisfy
A1.24)
>a,- p>0
9-n-i_
(b)
A1.25)
A1.26)
Proof, (a) Formula A1.24) is obtained as G.44) in the proof of Theorem 7.16:
because of A1.9) and A1.10) Щ intersects the right hand side of A1.24) in
a unique element. This element extends to a global section in H" and must
coincide with the restriction of v-,.
The formulas A1.7), A1.8), and A1.24) show
Kf{vl v»+w) € SU ¦ Э-" + ?C ¦ Э
*i
-*
A1.27)
On the other hand, КГ(Щ, Щ) = CU3,-1}} • Э^" (Theorem 10.7).
(b) One considers K^k\vi, vM+i_j) € Ом,о as apower series. The constant
term is K{fk)(v^ i^+1_,.) = hj ¦ ho because of Theorem 10.28 (i). The follow-
following calculation shows that the higher terms vanish. The third step uses A1.19)
once. Here X\,...,Xm € TM$ and k > 0.
(X,... Xm K(fk\vi, Vj))@) A1.28)
= Kf
m-l
e
/=0
D
Remark 11.4 In the case of a singularity with nondegenerate intersection form
the description of Kp in Theorem 10.13 gives a better idea why A1.26) holds:
Kp~k\vj, Vj) is by A0.47) the residue along {00} x M of a certain meromorphic
1-form on P1 x M. One can check that this residue is constant because of the
properties of v-, and Vj along {00} x M.

202
Frobenius manifolds for hypersurface singularities
In the case of a degenerate intersection form one can embed the generalized
Milnor fibration of the semiuniversal unfolding in a relative compactification
and obtain an analogon of Theorem 10.13 (cf. Remarks 10.14 (b) and 10.29
(a)). I checked that one can make a refinement of the choice (?/., y\) and that
one can also explain A1.26) in this case with a residue along {oo} x M, but
carrying out the details is quite intricate.
The section v\ plays a special role. It turns out to be a primitive form in the
sense of K. Saito. The germ v° 6 Щ has the principal part in C1. Then by
Theorem 10.33 the germ vi 6 П^ is represented by a volume form u(x, t)
dx0 ... dxn with м@) ф 0. By Corollary 10.8 (iv) it induces an injective period
map
v.%
M,0
x м- -v
¦V3>b
such that
Because of A1.19), v is then an isomorphism
Let
A1.29)
A1.30)
A1.31)
Si:=v-\Vi). A1.32)
By the Kodaira-Spencermap a : TMfi -* Oc,o from A0.3) we have the multi-
multiplication о on TM0 and the Euler field E = a~'([^])-
Lemma 11.5 concludes the proof of Theorem 11.1, except for the statement
that different choices (?/., y\) lead to different metrics g. This will be proved
after Corollary 11.6.
Lemma 11.5 (a) The 5,- satisfy S{ = e and [Si, Sj] = 0 and
A1.33)
-> SlFi0fromA0.38)
A1.34)
The Gmthendieck residue pairing JF on ?lF induces a flat metric g on M. The
vector fields St are flat with g(Si, <5M+w) = Sy. Let V* denote the Levi-Civita
connection of g.
(b) The composition ofv and of the projection r@).:
coincides with the isomorphism
Тм,о-+ nF,0 X ^ a(X) ¦ r
11.1 Construction of Frobenius manifolds
203
(c) The metric g is multiplication invariant and satisfies together with the
multiplication the potentiality condition.
(d) The Euler field E satisfies [S(, E] = A + «i - а,->5;.
Proof, (a) Crucial for (a) and (b) is formula A0.39),
r@)( „VxV^'uj) = a(X) ¦ r@)(w) A1.35)
for X 6 TM,0, со 6 Wf,0). For со = Vi and X = ^ it shows aE,) = 1, so
5] = e. Together with a(S,- о S,-) = aE,) • aEy-) it shows that A1.33) holds
modulo kerr@) = V^0' = П(о~1). Then A1.33) holds because of A1.19)
and A1.30). The vector fields 5,- commute because of
7-2.
= V(S; О Sj) - v(Sj oSi) = 0.
A1.36)
(b) The first statement is A1.35) for со = v\. The pairing K^ induces JF via
the projection r@) (Theorem 10.11). Then A1.26) shows g(Sh 5M+,_y) = Su.
The metric g is flat because the vector fields 6, commute.
(c) The metric g is multiplication invariant because JF satisfies the corre-
corresponding property A0.45). The manifold (M, o, e) is an F-manifold and e is
flat. In Theorem 2.15 it is proved that these three properties imply potentiality.
But the potentiality in the form V|Ey о Sk) = Vf.(<5; о Sk) also follows from
rewriting 0 = VKiJjlV^2u(l5Jt). Here one needs the OM,0-linear extension of
A1.33),
-VxV3-1l;(y)=:v(Xoy)-V3;1i;(V|r) A1.37)
for X, Y 6 TMfi.
(d) The vector fields E - ze and Эг + e and therefore also E + zdz are loga-
logarithmic along V by A0.9); in fact, E + zdz is also logarithmic along {oo} x M.
The sheaf Vg"'W@) = H(~l) has a logarithmic pole along V. Therefore
V?)V^1 Vi € V^Hi0)= z-Vi-v(Eo Si). A1.38)
Then A1.20) shows
~
= v(E о Si) + (a; + 1)V8~V
A1.39)
We write E = ?y SjSj for some ey e OMfi. Comparison of A1.39) and
shows
J J
Si{e}) - 0 for i ф j and 5j(e,) = 1 + ai - a,-.

204
Fmbenius manifolds for hypersurface singularities
Therefore V| E = [Sh E] = A + a, - a,)<5;.
One also now obtains the following nice formula as an Ом-linear extension
of A1.39). For X eTM
Z ¦ v(X) = v(E oX)+
+ a,)X -
A1.41)
a
Corollary 11.6 The period map v yields an isomorphism between the pairs
Gi@>, V) and (рг*Тм, V(~ ^) (the second one is a second structure connection
of the Frobenius manifold (M, о, е, Е, g)).
Proof. One compares (9.10) and (9.11) with A1.37) and (9.13) with A1.41).
One needs V - \ - § = V*E - f - | - § = V«? - B + at). D
Nowwecan prove the injectivity of themap {choices (U., yi)} -» {metrics g]
in Theorem 11.1. If one fixes U. then the construction of ы in A1.16) and
A1.17) shows that the metric g varies linearly with the element y\-
Suppose that two choices of (?/., y\) lead to the same metric. By
Corollary 11.6, there exists an automorphism of the flat cohomology bundle
which maps one extension 7^°> of its sheaf of sections to the other. The auto-
automorphism commutes with the whole monodromy group.
The monodromy group is generated by the Picard-Lef schetz transformations
of a distinguished basis of vanishing cycles. The automorphism commutes with
them and must map each distinguished cycle to a multiple of itself. The Coxeter-
Dynkin diagram is connected [La] [Ga]. Therefore the multiples are all the same,
and the automorphism is a multiple of the identity.
Thus the two extensions H<°) coincide and the two opposite filtrations coin-
coincide, because one can recover them from the extension at {oo} x M. Now the
elements y\ must also coincide. This finishes the proof of Theorem 11.1.
Remarks 11.7 (a) For most of the singularities there also exist metrics g other
than those constructed in Theorem 11.1 such that the F-manifold (M, о, е, Е)
together with g is a Frobenius manifold. Even the eigenvalues of Vs E are
different.
These other metrics arise in the same way as those in Theorem 11.1, but after
changing a very fundamental choice: One replaces the natural order < on the
set {a | e
~2nia
is an eigenvalue of the monodromy } U Z by another order
which satisfies G.9).
11.2 Deformed flat coordinates
205
Then the spaces H°°, C, V>~O0, and НЦ are still the same, but the V-
filtration, the spectral numbers of Яд', and the corresponding filtrations F. and
F#a'« from G.24) and G.27) usually change. Often opposite filtrations exist and
the whole construction in this section can be carried out.
(b) Let us sketch this for an example of M. Saito [SM3, 4.4]. The semi-
quasihomogeneous singularity / = x$ + x\ + x^x\ has spectral numbers
ot\ < ... < Q!25 which are as an unordered tuple (—1 + '+{+2 | 0 < i, j < 4).
One chooses st € C"*' as in A1.14) and A1.15) (this includes the choice of an
opposite filtration). It then turns out that v° — Sjfor/ > 2andu° = si+a-Va.S25
for some a e C\ with si e C~2'3 = C", УЭ;525 e C~x^ = C5.
If one now chooses a new order -< as in G.9) with —5^-5 then the spectral
numbers for this order are (no longer with indices fitting to their order) 5, = a,
for 2 < i < 24 and ci{ = a25 - 1 = -5, «25 = <*i + 1 = \
Now one can choose si = s,- for 2 < i < 24and?i =
Then v° = v? for 1 < i < 24 and v%5 =125-
The whole construction in this section can be carried out. One obtains a
Frobenius manifold with [5,-, E] = A + Si — a,)<5;.
(c) The different choices of opposite filtrations and of orders ~< seem to be
related both to certain transformations of Frobenius manifolds of Dubrovin
[Du3, Appendix B] [Du4, first part of Lecture 4] and to the Schlesinger trans-
transformations in [JMU][JM].
\-
11.2 Deformed flat coordinates
In this section we want to establish series of functions с\^ е Ом,о which
are implicit in the construction of section 11.1 and which are very close to
Dubrovin's deformed flat coordinates. These deformed flat coordinates play a
major role in many of Dubrovin's papers on Frobenius manifolds. They lead
to rich hidden structures [Du3, Lecture 6], [DuZl], [DuZ2], [Gi8], whose
meaning for singularities has still to be explored.
We will use our series of functions in chapter 12 in order to establish a
canonical complex structure on the /x-constant stratum and study a period map.
Remark 11.8 The most detailed description of the deformed flat coordinates
can be found in [Du4, Lecture 2]. We refrain from repeating that here and
restrict ourselves to some comments.
The first structure connection У(~^ (Definition 9.6) on the lift pr*TM of the
tangent bundle to P1 x M is flat and has a logarithmic pole along {0} x M. The
same holds for the induced connection on the dual bundle pr*T*M.

206
Frobenius manifolds for hypersurface singularities
Dubrovin's deformed flat coordinates %(t, z) are multivalued functions on
C* x M whose restrictions to slices {z} x M form locally flat coordinates with
respect to V(~2>.
They also have the best possible behaviour with respect to dz. To make that
precise, let dM denote the differential only with respect to the coordinates on M,
not with respect to z. Then the differentials &M% are multivalued flat sections
in pr*T*M |C* x M with respect to the induced connection.
The deformed flat coordinates Ц are usually written as a tuple
, z) 7m(t, z)) = @i(r, z),.. •, em(t,
я A1.42)
[Du4, B.83)], [DuZl, B.63)], [DuZ2, C.13)]. The matrices /i and R are sup-
supposed to be chosen such that the following holds:
The functions 0,-(f, z) are holomorphic in С* х М. Their differentials d^fy
form a basis of elementary sections in pr*T*M |C* x M. (Here ±д is the
matrix for the (by assumption) semisimple residue endomorphism, the matrix
R carries the Jordan block structure of the monodromy.)
The coefficients 9jp(t) in the expansions Oj(t, z) = J2p>o^ip^z>> are more
or less equivalent to the coefficients c,y+ below. 'More or less', because our
coefficients come from the second structure connection V(~5> which is close to,
but in general not equal to the Fourier dual of V(~5> (cf. Remark 9.9 (b)). The
9jP are the real starting point for the profound structures in [Du3, Lecture 6],
[DuZl], [DuZ2], Gi8
We consider the same situation as in section 11.1 and use the notions es-
established there. Most important are the global sections u,- and the elementary
sections si. The germs in H@>@0|0) of the global sections v, will be developed
in series
1<У</г p>0
with coefficients c]f € Ом,о- Because of A1.12) and A1.14) one has
A1.44)
where? = К Therefore the germs v,- in (oo, 0) can be written uniquely as sums
of elementary sections ~zpSj. But we want to write them with (—Vs. )pSj.
For a $ Z<o there is no problem. The sheaf W' has a logarithmic pole
along {oo} x M, and — dz = ?(?%). Therefore the sections (—Vaz)pSj with
ay — p = a and p > 0 form a basis of the same subspace of С as the sections
"zpSj. But for а е Z<o there are two related problems:
77.2 Deformed flat coordinates
207
(I) The sections (—V3.)''sy with a,- — p = a and p > 0 generate a strictly
smaller subspace of С than the sections ~zpSj. A priori it is not clear that
the elementary part of d,- in Ca is in this smaller subspace. But it will turn
out to be the case.
(II) (,-^b.ysj — 0 for certain j and p. The coefficients с(У in A1.43) for these
j and p are not unique, but arbitrary.
In order to obtain unique coefficients с? we have to introduce a dummy expo-
exponent ? e N and consider the analogon of A1.43) for (—Va. )^ ы for all ? e N.
That shifts the orders of the elementary sections out of the bad domain Z<0.
Each coefficient c\f will be uniquely defined by the expansion of (-V3.)~^u,
for sufficiently large ?. The same trick will be used to establish the relations
between the c\j\ It is clumsy, but worth the effort. With the first structure
connections, one would not have this difficulty.
Of course, (—V3z)~^D; is the section in W@'|C x M which is welldefined
by the proof of Theorem 10.7. The germ at (oo,0) is a germ in Ц~$ ¦
Theorem 11.9 (a) Let u,- and s-, be as in section 11.1. There exist unique
coefficients cff e Ом,о {p € Z, i, j e {1,..., fi}) such that the germs at
(oo, 0) satisfy
(-У3Г^= Y.Y.cll?(-Va,)"-llsj (П.45)
Om,q be
A1.46)
A1.47)
for all ? e N. The coefficients satisfy c]f = Ofor p < 0 and cfj} = 50
(b) Let Si = ir'(i>i) be the flat vector fields from A1.32). Let a\} €
the multiplication coefficients with Si о Sj = ^^ af-S^. Then
and
C
Especially Si = e, akjx = <5,ь and
«,#=4.
A1.48)
A1.49)
A1.50)
The last equation says that r,- := c(,)' are flat coordinates on M with -^ = 5,-.

208 Frobenius manifolds for hypersurface singularities
(c) For any i,je[l,...,n},re Z,
s,-M+waOr= ? B-^M
p,q: p+q=r к
The right hand side has for r > 1 the linear terms (—l
for r > 2 additional quadratic terms. Especially
С
с;
V
C(D
(d) Let nij eCbe the coefficients with (cf. A1.6))
Then nij = Oforoij — 1 — а,- ф 0 and
Z-Vi = (a,- + l)V^'v; +^W,7V; ¦
and for p > 2
1_у + с;д+1_,. anrf
A1.52)
A1.53)
a,)t»y, A1.54)
(П-55)
77ге Euler field E satisfies
EoSi = J2n>JSJ ~ ?cU){aJ
E =
A1.58)
Proof, (a) Because of u,- = v| (V^*v,), the elementary part of v,- in C" is
contained in v| Ca+$ for any | e N. The same holds for the elementary parts
of (—V8i)~?v,. This solves the problem (I) before Theorem 11.9. For each f
one has an expansion A1.45). The coefficients c)j are independent of ? and
uniquely determined by the expansions for sufficiently large ?. Now с? = О
for p < 0 is trivial and cf^ = 5y follows from A1.18).
(b) Formula A1.46) is A1.33) rewritten. One complements (—Vg7)~? on both
sides and inserts the expansions A1.45). One obtains A1.47) by comparison of
the coefficients of (— V3j)p~Ss, for sufficiently large f. The rest is obvious.
11.2 Deformed flat coordinates
209
(c) First we formulate an approach which does not work, but which helps
to understand A1.51). One would like to ignore that certain (— Va )pSk
vanish. One would like to extend the pairing Kf from section 10.6 Ом off ЭГ1}}
[3z]-sesquilinear and calculate Kp(Vi, Vj) with this extension of Kf and with
the expansions A1.45). One would obtain A1.51) because of A1.15) and
A1.26).
The real proof is somewhat similar to the proof of A1.26). One has to con-
consider the right hand side of A1.51) as a power series in OMfi and discuss the
coefficients separately.
Let Xb ..., Xm e TM,o (the empty set for m = 0), r > 0 and f > r + m.
Then the number
8mO8ill+i-j8ro ={Xl...Xm 4r?)(( -V,,)*^, ( -V3ir2S)))@)
A1.59)
can be calculated as follows: One inserts the expansion A1.45), uses Theorem
10.11 (v), Theorem 10.28 (i), A1.15), and the properties of Kf. One obtains
the value at 0 of the derivative by Xi... Xm of the right hand side of A1.51).
Details are left to the reader.
(d) Formulas A1.54) and A1.55) will be proved simultaneously by the follow-
following calculation. The first step uses A1.45), A1.53) and [г, (-Эг)*] = *(-Эг)*-'.
In the second step the expansion A1.45) is applied to the terms for p = 0, in
the third step to a part of the terms for p — 1.
j.p
j.k p>\
? ? (c

210
Frobenius manifolds for hypersurface singularities
-p-
к р>2
The germ at (oo, 0) of z • v,- is contained in
V (п.60)
A1.61)
The restriction toCxM of г-v,-is containedin7^(°>|CxAf. Therefore z-n; is con-
contained in ф;. Om,o ¦ Vj Фф; Одт.о • Va~' vj. The difference between (-V8. )~*z¦ v,
and the first two lines of A1.60) is an element of (-У3.Г*(ф; Ом~о ¦ vj ф
ф;- О мл ¦ Vg^ Vj). If it were nonvanishing then the expansion into elementary
parts would yield a nonvanishing part in (-^Зг)~?(ф. Ом,о ¦ Sj Ф ф ¦ Ом,о •
V^sj). But the sum of the last three lines of A1.60) does not have such a part.
Therefore the difference and this sum are both vanishing. This gives A1.54)
and A1.55).
(e) Formulas A1.56) and A1.57) follow from A1.39) and A1.54). Again
with A1.39) one obtains a,v; = V?+z3jv,. One puts into both sides the expan-
expansion A1.45) and applies A1.53) on the right. Comparison of coefficients gives
A1.58). n
Remarks 11.10 (a) There is a rich structure hidden in the equations in
Theorem 11.9, as is clear from Dubrovin's and Zhang's work. The most impor-
important coefficients are the c(,f in view of A1.48). Central for Dubrovin is equa-
equation A1.47), usually written as a second order equations for the c<f+1). For the
/г-constant stratum we will mainly regard the equations in (d). Equation A1.55)
shows that the coefficients c\]j> and the c\f with otj - p - a,- = 0 determine all
coefficients c^\
(b) For fixed coefficients c\f e С with c\f = 0 for a, - p - щ < 0
the equations A1.55) and A1.52) imply A1.51). This was proved in [He4,
Proposition 5.5]. It was used to describe coordinates for the fibres of the clas-
classifying space DBL for Brieskorn lattices as a bundle over a classifying space
Dpmhs for polarized mixed Hodge structures.
But if only c\f -OfoTUj-p-at < 0 then the coefficients with aj -p -a,- =
0 satisfy equations of the type A1.51) which do not follow from A1 55) and
A1.52).
11.3 Remarks on mirror symmetry
211
11.3 Remarks on mirror symmetry
Dubrovin's definition [Dul][Du3] of Frobenius manifolds formalized a part of
the structures which the physicists Witten [Wit], Dijkgraaf, E. Verlinde, and H.
Verlinde [DVV] had found studying (moduli spaces of) topological field theories.
Different ways in physics to establish such structures lead to phenomena
which are now comprised in the famous notion mirror symmetry.
There are several very different versions of mirror symmetry. A vague para-
paraphrase of one version is that one has two sides, the A-side and the B-side: on
the A-side one looks at data related to the Kahler geometry of a manifold X,
on the B-side one looks at data related to the complex geometry of a family of
manifolds Y,. These data should be isomorphic if the manifold on the A-side
and the family of manifolds on the B-side are mirror dual to one another. See
[Vo] for a detailed discussion and references.
The structure on the A-side of this version of mirror symmetry comes from
genus 0 Gromov-Witten invariants and is called quantum cohomology . Within
the frame of algebraic geometry it has been established by Kontsevich, Manin,
and many others (see [KM], [Man2], and references therein). If X С VN
is a manifold with Hodd(X, C) = 0, then its quantum cohomology can be
encoded in a formal germ of a Frobenius manifold, where the formal germ of a
manifold is (H*(X, C), 0). The part of the structure which lives in some sense
on (H2(X, C), 0) С (Н*(Х, С), 0) is called small quantum cohomology (see
[Man2] for any details).
The structure on the B-side of this version of mirror symmetry is related to
period integrals, Picard-Fuchs equations, hypergeometric functions. If Y, is a
family mirror dual to X, then the parameter t should be in a space isomorphic to
a neighbourhood of 0 in H2{X, C), and the data from Y, should be isomorphic
to the small quantum cohomology of X. The firstexample of this was proposed
be Candelas, de la Ossa, Green, Parkes [CDGP]. It was proved together with
many other cases by Givental [Gi4].
In view of the full quantum cohomology and the Frobenius manifold on the
A-side, one may ask about an extension of the family Y, and the data on the
B-side, such that in that case one also obtains a Frobenius manifold, and such
that it is isomorphic to the Frobenius manifold on the A-side. That would be a
stronger version of mirror symmetry.
One approach to this is provided by the Barannikov-Kontsevich construc-
construction [BaK][Bal][Ba2]. Using tools from formal deformation theory, there an
extended moduli space M (a priori a formal germ, in good cases a man-
manifold) is constructed such that a subspace Mcs (cs for complex structure)
parameterizes a family Y, of complex manifolds. Under certain assumptions

212
Frobenius manifolds for hypersurface singularities
О
w
the space M. is equipped with the structure of a Frobenius manifold. Build-
Building on Givental's results, Barannikov proved isomorphy of this Frobenius
manifold with that from quantum cohomology of X when X is a projec-
tive complete intersection Calabi-Yau manifold [Bal]. A major and still
unsolved problem is to find objects which are parameterized by points
t e M - Mcs.
Another approach to get Frobenius manifolds on the B-side is motivated by a
proposal of Givental [Gi3] and Eguchi, Hori, Xiong [EHX]: if X on the A-side
is not a Calabi-Yau manifold, but, for example, a Fano manifold, one should
consider on the B-side as mirror dual a family (У,, /,), where /, : Yt —> С is
a function on Y,. If X is Calabi-Yau then /, should be constant, but in other
cases it may be a function with isolated singularities on the (now not compact,
but usually affine) manifold Yt.
Then the data on the B-side for the family (Y,, /,) come from oscillating
integrals. Givental proved in many cases that they are equivalent to the small
quantum cohomology of the mirror dual manifold X [Gi3][Gi4][Gi5][Gi6]. In
many cases the parameter space of the family is isomorphic to an open domain
in H2(X, C), and at all isolated singularities of the single functions /(, the
family is a /x-constant deformation.
Now the extension of the family (Yt, /,) to a family with parameter space
M with dim M = dim H*(X, C) is often not difficult: one has to consider also
deformations of the functions /, which are not д-constant at the singular points.
One arrives at a situation which generalizes the semiuniversal unfolding of a
local singularity / : (C"+1, 0) -* (C, 0).
Sabbah saw that one can often establish the structure of a Frobenius man-
manifold by a procedure which is similar to that in section 11.1. But one needs
oscillating integrals instead of the GauB-Manin connection. This is because
the middle cohomology group of the smooth fibres f,~i(z) usually has a di-
dimension Ф dim M. Also, the results on the existence of a Frobenius manifold
are not as complete as in the case of a local singularity. This is discussed in
[Sab2][Sab4]. The best positive results are due to him [Sab3][NS]. We give
a sketch of our understanding of the situation and part of his work in the next
section.
The family of functions mirror dual to P" was proposed and studied in [Gi4]
and [EHX]. The construction of its Frobenius manifold was carried out by
Barannikov [Ba3]. It fits into the frame given by Sabbah.
11.4 Remarks on oscillating integrals
The version of mirror symmetry which was discussed in section 11.3 asks in its
strongest form whether certain Frobenius manifolds from quantum cohomology
11.4 Remarks on oscillating integrals
213
of projective manifolds are isomorphic to Frobenius manifolds coming from
suitable families of functions on suitable manifolds.
Semiuniversal unfoldings of local singularities are not suitable families of
functions, simply because their spectral numbers never fit to those from quantum
cohomology.
But one may hope to find suitable families of functions starting from func-
functions which were studied by Sabbah [Sab2][NS]: M-tame functions on affine
manifolds (definition at the end of this section). Sabbah indicated how one may
proceed to construct Frobenius manifolds from them [Sab3][Sab4].
The construction is similar to that in section 11.1. But instead of the GauB-
Manin connection and a second structure connection one considers oscillating
integrals and a first structure connection. The two most difficult steps are:
(a) to construct a meromorphic connection on a trivial vector bundle over
P1 x M which looks like a first structure connection;
(b) to find a certain global section (a. primitive form) which induces an iso-
isomorphism between the lifted tangent bundle n*TM and this trivial bundle.
Step (a) involves (the spirit of) oscillating integrals and the choice of an
opposite filtration. By some work of Sabbah, it can be carried out for a large
class of functions, all M-tame functions on affine manifolds. But the existence
of a primitive form in step (b) is for the moment only clear for a much smaller
class of functions, for polynomials with nondegenerate Newton boundary.
In [Ba3], Barannikov treated (independently) the family of functions mirror
dual to Pn+1. Although the presentation is completely different, this fits into
Sabbah's frame.
Not all steps in Sabbah's programme have been worked out in detail. And
we will not do it here. We will restrict ourselves to formulate this programme
in 5 steps and then comment upon some aspects of the steps.
Step 1: Construct a family of functions /, : X, -> A, t € M С О\ with
the following properties.
(a) The set A = {z e С | \z\ < »?} is a disc, X, С CN is a Stein manifold of
dimension и + 1, X := \Jt X, x {t} С CN x M is a manifold, M is a (small)
open subset of C*.
(b) The function/, : X, -* Д has isolated singularities with д = Z^eSingC/,)
m(/(. x), and for any 7 e M the restriction of the family of functions to the
multigerm at Sing(fy) x {?} С X is a product of semiuniversal unfoldings of
the singularities of jj.
(c) The map
<p : X -*¦ A x M
(jc, О м- (/,(*), 0
A1.62)

214
Frobenius manifolds for hypersurface singularities
11.4 Remarks on oscillating integrals
215
fir.
is a C00-locally trivial fibration outside of a discriminant V С А x M. For
any (г, t) e V, the fibre <p~l(z, t) = /,~"'(z) x {t} is singular, and there exist
arbitrarily small neighbourhoods U\ С X of Sing(cp~l(z, t)), (/2CA of г, and
U-i С Moffsuchthatthe restriction of ipto^~'((/2 x (/з)-(/[ is a C°°-locally
trivial fibration.
Step 2: Construct a flat bundle Ни/ of rank д over С* х М (definition in
A1.66)) with connection VLef such that the fibres of the dual bundle can be
interpreted as spaces of Lefschetz thimbles. Let Иье/ be the sheaf of holo-
morphic sections of #z.e/. Define a certain pairing on the sheaf n^Tiuf (here
я : С* x M ->¦ M).
Step 3: Extend the sheaf TiLef to a free 0CxM-module 1iflf such that the
pair CW^e/» VLef) has a pole of Poincare rank 1 along {0} x M. This uses
the GauB-Manin connection of <p and a Fourier-Laplace transformation with
parameters. A geometric interpretation imitates oscillating integrals on a
(co)homological level.
Step 4: Try to extend the sheaf fiflf to a free Or\ xW-module Hflf with a
logarithmic pole along {oo} x M. This is aBirkhoff problem with parameters.
Sabbah showed that this is solvable for similar reasons as in the case of a
local singularity if one starts with an M-tame function on an affine manifold.
He established a mixed Hodge structure on the space Hfy of manyvalued
flat global sections of Huj [NS]. An opposite filtration to its Hodge filtration
induces a solution of the Birkhoff problem.
Step 5: Try to find a global section V\ (a primitive form) in Hflf which
is flat at {oo} x M with respect to the residual connection there, which is an
eigenvector of the residue endomorphism at {oo} x M, and which induces an
isomorphism Тм —*¦ л^И^ by a period map. Then a part of the pairing in
step 2 induces a flat metric in TM such that M gets a Frobenius manifold. This
isomorphism also identifies (Ti-flf, VLef) and the first structure connection
(V(~5~!), рг*Тм) (with the coordinate г = — J-, whereЛ is a coordinate on C*
used below).
Because of condition (b) in step 1, M will be a massive F-manifold with
Euler field and smooth analytic spectrum.
The cohomology bundle of the fibration <p from A1.62) is
Hn:= У Hn(fr\z)X). A1.63)
(г,/)еДхЛ/-Й
If the spaces X, are not contractible then in general the fibres of Hn do not have
dimension д. In that case there is no chance of extending the GauB-Manin
connection to something isomorphic to a second structure connection. But by
condition (c) in step 1, the spaces of Lefschetz thimbles have the right dimension
д. This is the reason why oscillating integrals will be used. To define Lefschetz
thimbles, fix for a moment# e C* and t e M. For sufficiently small ? > 0 let
Д<Л) := {z € A | \z — г]щ\ < e} be a neighbourhood in Д of the boundary
point т)щ. Then X\K) := f~l(A{n)) is homotopy equivalent to a smooth fibre
off.
A Lefschetz thimble in X, with boundary in x\ is an (n + l)-cycle Г =
Цехао,]]M(г)> where У : [°> !] -> л is a Path with K@) 6 f(Sing(f,)),
K(@, D) П f,{Sing{f)) = 0, K(D 6 A№), and S(z) for z e y([0, 1]) is a
continuous family of n-cycles in /r~'(z) which vanish for z —»¦ y@).
Condition (c) in step 1 and standard arguments (e.g. [AGV2, ch. 2], [Lo2,
E.11)]) show that one obtains a space homotopy equivalent to X, if one glues
ix Lefschetz thimbles to xf\ (First one chooses t e M such that f has д
A [-singularities with different critical values and considers Lefschetz thimbles
over ц. nonintersecting paths from f(Sing(f,)) to Д<л).) Therefore
«
A1.64)
and this space is generated by the relative homology classes of Lefschetz thim-
thimbles. There is an exact sequence
0 -+ Hn+l(X,, Z) -> Hn+l(Xt, Xf\ Z) A1.65)
^ Hn(X?\l)-* Hn(X,,Z) ^ 0.
The bundle Hief in step 2 is
HLef.= U Нп+1(Х„Х?\СУ. A1.66)
(Я,;)еС*хМ
A pairing on ж» 7^/ can be defined using the intersection form for Lefschetz
thimbles, which is a perfect pairing
< .,. >: Hn+1(Xt, Xf\ Z) x Н„+1(Х„ Х\-к\ Z) -> Z. A1.67)
It induces a pairing <.,.>* on the dual spaces. Then
KLef : n^Hief X ntHief -»¦ Л*ОсхМ
(а,Ь)\-*(Н\-><а(П),Ь(-П)>*) A1.68)
is Ом -bilinear and sesquilinear тй.
The definition of the extension Tlflf of H-uf to С х М in step 3 is not easy.
One has to start with an extension 7Y@) to Д x M of the sheaf of sections of the
cohomology bundle Hn. This should be defined using relative (n + l)-forms as

ел;
216
Frobenius manifolds for hypersurface singularities
in chapter 10, but the definition in Lemma 10.2 probably gives the right object
also in this more general situation. Then a Fourier-Laplace transformation with
parameters (i.e. keeping t and 3,, changing z ь> —3r, dz ь> r, where r := |)
should give H(°lf.
The best way to handle this has still to be fixed. A geometric interpretation
looks roughly as follows. One extends the homology bundle and the cohomol-
ogy bundle H" of cp to flat bundles over С x M - T>. If Г = LUy([o,i])S^
is a Lefschetz thimble as above, one extends the family [S(z)] of homology
classes of cycles to a continuous family in the extended homology bundle over
y([0, oo)), where y([l, oo)) is the half line which starts at y(l) and goes in the
direction -щ. If со е 7r*7^@) now extends to a global section over CxM with
moderate growth along {oo} x M, one defines
[е-г*а>](\Г]):=
[
y([0,oo))
n co{[8{z)\)iz.
A1.69)
This imitates an oscillating integral. One has to show that it gives a holomorphic
section [e~z^co] in HLef. Then such sections generate Hflf as an OcxM-module
As soon as one has proved formulas which establish the Fourier-Laplace
correspondence between sections of 7^@) and H\2/ (i-e- г и- -Эг,Эг к t,
where т = |), it is not hard to see that T~L\jef is a free Осхм-module with a
pole of Poincare rank 1 along {0} x M.
If the family /, : Xt —*¦ Л is a semiuniversal unfolding of a local singularity
then (—1);lt~ BgL+i Kief on itt'Hflf is the Fourier dual of K. Saito's higher
residue pairings Kp on 7r*7^<0) from section 10.4. This follows from [Ph5,2eme
partie 4].
In order to find an extension to a free Op \xM -module T-6?lf with a logarithmic
pole along {oo} x M, one can restrict to a fixed parameter t e M. Let n"LL\t
be the free Ос-module of restrictions of sections in Tiflf to Hief\cx(i)- It has
a pole of order < 2 at 0. One wants to solve the Birkhoff problem for T^ilffc
in the spirit of sections 7.2 and 7.4, by defining a filtration F* in H?°f from
7^j^|7 and choosing an opposite filtration.
But the procedures in sections 7.2 and 7.4 do not apply, because H*j%f \r has a
pole of order < 2 at 0. Sabbah found a very nice variation of these procedures.
One has to look at global sections in H^ \j with moderate growth at oo and has
to define principal parts for them using the V-filtration at oo. The principal parts
induce a filtration F' on Я^-. Any opposite filtration gives rise to a solution
of the Birkhoff problem for nfef \j. See [Sab4, III 2.b and IV 5.b] and [Sab2]
for details.
11.4 Remarks on oscillating integrals
217
A profound result of Sabbah is that F* is a Hodge filtration of a mixed Hodge
structure if f~t : X~t —> Д is the restriction of an M-tame function /7 : Y —> С
on an affine manifold К DI7 [NS][Sab2]. Then opposite filtrations exist and
the Birkhoff problem is solvable.
Definition 11.11 ([NZ][NS]) The function /7 : Y -+ С is M-tame if for some
closed embedding Y с <CN of the affine manifold Y the following holds: for any
r] > 0 there exists an R(n) > 0 such that the spheres S™?1 = {x € CN | \x\ =
R(r])} are transversal to ffl(z) if г € Д.
Then the restriction /7 : B^|' П ffl(A) -*¦ Д is the analogon of a Milnor
fibration of a local singularity, if Д contains all critical values of /7 : Y -»• C.
In [Sab2] a related more algebraic notion, cohomologically tame, is considered.
But it is not clear whether it implies M-tame. Suppose now that /7 : Y -> С is
M-tame.
Then one can realize the definition of W^fr and the formula A1.69) by
proper oscillating integrals over Lefschetz thimbles in the sense of [Ph4] [Ph5].
One can also realize the family /, : X, -»• Д with t € M by global functions
/, : Y -> С But in general these functions will not be tame in any sense
and will have nonisolated singularities or too many isolated singularities. In
order to obtain a family of functions as in step 2, in general one has to cut out
bounded spaces X, С Y, which contain the singularities of ft, but exclude the
bad phenomena coming from infinity for t Ф t.
In step 5, the existence of a primitive form is unfortunately at the moment only
clear for a much smaller class of functions, for polynomials /7 : Cn+1 —> С
with nondegenerate Newton boundary [Sab3][Sab4]. The problem is that the
analogon of M. Saito's Theorem 10.33, that the smallest spectral number «i
has multiplicity 1 and corresponds to volume forms, is not clear in the case of
globalfunctions f?: Y -> C. But it is not difficult to establish this in concrete
cases, and it may well hold for many classes of functions important for mirror
symmetry.

о
Chapter 12
д-constant stratum
Two applications of the results of chapter 11 are given here. In section 12.1 a
canonical complex stracture on the д-constant stratum of a singularity is es-
established. The most difficult part is to prove that it is independent of the choice
(U., у\) in Theorem 11.1. In section 12.2 the period map from ад-constant stra-
stratum to a classifying space for Brieskorn lattices from [Hel][He2][He3][He4]
is taken up. It is shown that the map is an embedding. This strengthens a result
of M. Saito.
12.1 Canonical complex structure
The д-constant stratum of a singularity can be equipped with a canonical com-
complex structure. This will follow from the construction of Frobenius manifolds in
chapter 11 and from Varchenko's result that the spectral numbers are constant
within a д-constant family (Theorem 10.32, [Va3]). One can write down the
complex stracture quite explicitly for a choice of an opposite filtration. But the
proof that it is independent of this choice is technical.
Let / : (C"+1,0) -* (C, 0) be a holomorphic function germ with an isolated
singularity at 0, with Milnor number д, and with n > 1. We choose a representa-
representative F : X -*¦ Д of a semiuniversal unfolding F: (C"+1 x CM, 0) -*¦ (C, 0) as in
section 10.1, that means, Д = B\ сС,М = В,"сС',Л'= F-\A)n(B^+i x
M) for s, S, 9 > 0 sufficiently small. The function F, : X П {Bne+X x {t}) -»• Д
is the restrictionto the parameter t = (t\,..., fM) € M.
The д-constant stratum 5M с М is
S^={t €M \ Ft has a singularity with Milnor number д and critical value 0}.
A2.1)
It is a subset of the discriminant V С М,
V = {t e M | F, has a singularity with critical value 0}. A2.2)
218
12.1 Canonical complex structure
219
Both can be described using the canonical stracture of M as an F-manifold
{M, о, е, Е) (see section 10.1 and section 5.1 for the definition of this structure).
The discriminant is V = det(W)~'@), where U is the multiplication with the
Euler field.
Gabrielov [Ga], Lazzeri [La], and Le [Le] proved that F, has only one
singularity if all its singularities are concentrated in one fibre. This is used in
the following characterization of 5Д.
Theorem 12.1 (a) The ^-constant stratum 5M С М is an analytic subvariety.
It is
5Д = V П [t € M | (T,M, o, e) is irreducible} A2.3)
= {teM\U: T,M -»¦ T,M is nilpotent} A2.4)
= {t € V | the e-orbit through t intersectsV only in t]. A2.5)
(b) The germ ((M, 0), o, e, E) of an F-manifold together with the germs
(U, 0) and EM, 0) is independent of the choice ofF, it is unique up to canonical
isomorphism.
Proof, (a) Equation A2.3) holds because (T,M, o, e) is isomorphic to the sum
of the Jacobi algebras of the singularities of F,. Both sets on the right are analytic
(cf. Proposition 2.5 for the second). So SM is analytic. The endomorphism U
corresponds to the multiplication by F, on the Jacobi algebras. Equations A2.4)
and A2.5) follow with the result of Gabrielov, Lazzeri and Le.
(b) Theorem 5.4. ?
Now we consider the construction in chapter 11. We will quite freely make
use of the notations introduced there. The choice of an opposite filtration U.
and a vector y\ as in Theorem 11.1 and the choice of s\,..., s^ as in A1.14)-
A1.16) induces a flat metric g on M, a basis 5i,..., 5M of flat vector fields, and
coefficients c\f (Theorem 11.9).
Theorem 12.2 Let (U., y\), s\, ¦ ¦ ¦, V S\, ¦ ¦ • > V andc\f be as in chapter 11.
Define coefficients e,y 6 Ом,о by E о &, = ]TV SijSj.
Then there is an equality of ideals
(ey I a} - 1 - a,- < 0) = (c<]> | aj - 1 - a, < 0) A2.6)
= (cff \otj-p- a,- < 0) С Om,o.
This ideal defines the /г-constant stratum EM, 0) С (М, 0).

220
fi-constant stratum
Proof. In A1.56) nij = 0 for otj - 1 - а,- ф 0. This shows s,j = -(a,- - 1 -
oti)c]V for ay - 1 - a,- ^ 0 and the first equality. The second equality follows
with A1.55). We have to show
M = [t 6 M | c\)\t) = 0forotj-\-a,- < 0}.
A2.7)
The matrix (e,j(r)) for r as on the right hand side is nilpotent. Equation A2.4)
gives the inclusion Э . For the opposite inclusion С we need Varchenko's
result that the spectral numbers are constant within a д-constant family
(Theorem 10.32, [Va3]).
Fix a parameter t e S^. The restriction to (Д x {r}, @, t)) of the sections
of TC^ gives the Brieskorn lattice H([(t) of the unique singularity of F, (see
Remark 12.3). Its spectral numbers are ab ..., aM by Theorem 10.32. Let
v\ € Hg(t) be the restriction of the global section u,- to this Brieskorn lattice.
Consider the expansion A1.45) for v't. It is finite, c^\t) = 0 for a,- - p < c*i,
because of Я^'@ С V".
The inclusion С in A2.7) is equivalent to the claim v? 6 V"'. This holds
obviously for v i. If any v\ with i > 2 had a principal part of order < otj then
this would be independent of the principal part of v\ because that contains s\
and v\ with i > 2 does not contain s\. The second spectral number would be
smaller than аг, a contradiction. Therefore v\ e V1 for i > 2. The claim
follows inductively. D
Remarks 12.3 The identification of the restrictions to (Д x {t}, @, r)) of the
sections of Tim with the sections in the Brieskorn lattice Щ'A) of F, for t e S^
is not as self-evident as it looks.
The restriction to (Д x {t}, @, t)) of the fibration ip : X -*¦ A x M, (x,t)i-*-
(F(x, t), t) of the unfolding F is not necessarily a Milnor fibration for the unique
singularity of F,. It could happen that one has to choose e(t) <? s to make sure
that all balls in C"+1 which have radius s' < e(r) and are centred at the singular
point of ^,"'@) intersect F~l(Q) transversally.
But according to [LeR, Lemma2.2] the restriction cp\(A x {t}, @, r)) is fibre-
wise homotopy equivalent to a Milnor fibration for F,. The cohomology bundles
are canonically isomorphic. The restrictions to (Д x {t}, @, t)) of sections of
Нф) give the Brieskorn lattice ЩЦ) of Ft.
Also, the space of multivalued flat global sections of the cohomology bundle
of F, is canonically isomorphic to H°°. The spaces of elementary sections for
this cohomology bundle can be identified with the spaces Ca of elementary
sections near {oo} x M for the extended cohomology bundle of <p. Compare
the observations A)-(Ш) before Lemma 11.2.
72.7 Canonical complex structure
221
This allows the Brieskorn lattices H^t) for all t e 5M to be considered as
sublattices of the same lattice V >-1. In section 12.2 we will consider the period
map SM ->• DBL to a classifying space for such lattices.
Theorem 12.4 The ideal in A2.6) is independent of the choices of su ..., sM
and (U.,y\).It defines a canonical complex structure on the [i-constant stratum
Proof: 1st step: We fix an opposite filtration U.. That does not induce a unique
metric, but a unique flat structure on M. Then adE acts semisimple on the
space of flat vector fields with eigenvalues -A + ai - ctj). Let 5i,..., 8^ be
any basis of flat vector fields with [5,, E] — A +c*i - a,-M,-. Define?,^ € OMfi
by E о % = J\ еи ¦ 1j. Then the ideal (eu \ ot} - 1 - a,- < 0) is the same for
all such choices of 1\,..., \, because a base change mixes only % with the
same orders a, and ay. Therefore the ideal in A2.6) depends at most on (/., not
on y\,s\, ...,jm.
2nd step: Let U. and U. be two opposite nitrations. They induce subspaces G"
and Ga of Ca as in A1.5). They satisfy
and vice versa. Choose sections ji, ... ,5^ as in A1.14) and A1.15) for U.
There exist unique sections 5,- 6 G°" of the form
Si=
with df = Sij. They satisfy A1.15). Let v,- and U, be the corresponding global
sections for the two extensions of Hi0) by U. and U.. Over С х М they
are bases of the same sheaf, along (oo)xM they generate different exten-
extensions with logarithmic pole along {00} x M, but both with spectral numbers
-ai,..., —aK € (-n, 1). Therefore the coefficients in
й = фв,,(г,О-«/ 02.9)
satisfy
A2.10)
m=0

X
Xr'
222 \i-constant stratum
Because of A1.54) one can rewrite this with Vr1,
A2.11)
with unique ^т) 6 dM.
Now one can calculate the coefficients c^ in terms of the c-?\ One uses
A1.45) for и,- and for u,-, A2.11) and A2.8). One obtains
l
jtk,m,p,q:
ij Cjk "kl
A2.12)
Here note that b^ = 0 for m < -n and for m > 0; dff = 0 for q > 0 and for
_. _ _. / (\. j\") __ a . \p) f\ i?_ a @) p
"/ "~ 4 ~ ak r U, "W = "kit Cjk = U 1ОГ /J < (J; Сд ^ од.
We want to show
c^ e (сд' | c^ - p - aj < 0) for a/ - r - a,- < 0.
Because of A2.12) it is sufficient to show the following claim:
bV 6 (c<fk I ak ~ P ~ <*j < 0) fora;-r-a,- < 0. A2.13)
Claim A2.13) will be proved inductively over the size of щ — r - a,. Fix fi < 0
and suppose that A2.13) is true for a, - r - a,- < p. Fix a triple (i, /, r) with
a; - r - a,- = 0 and r < 0. Then c^' = 0. Equation A2.12) gives a relation for
the monomials on the right hand side. Modulo the ideal (c^ | ctk — p — ctj < 0)
the only monomials left are those with
p > 0, ak - p - oij = 0, otj -m-a; =
and those with
A2.14)
A2.15)
Because of A1.24) c)pk\<X) = 0 for p > 0, ak - p - a,- = 0.
/2.7 Canonical complex structure
223
So, A2.12) modulo the ideal is a linear relation between ?>-m) with aj ~m —
a, = P; the only eventually invertible coefficients are 1 for bff and d^~m) for
fcjj' with m > r.
Putting together the relations A2.12) for all triples (i,l,r) as above, the
coefficient matrix is triangular and unipotent for t = 0. Therefore the claim
holds for (i, I, r) as above. This finishes the induction and the whole proof.
?
The Kodaira-Spencer map yields a canonical isomorphism (cf. A0.1))
Э [dF I
a0 : T0M -> Oc»+,,o/7/, — ^ — A2.16)
°ti I °fi (=0J
between the tangent space at 0 and the Jacobi algebra O/Jf. The space
л
A2.17)
(cf. A0.87)) is a free module of rank 1 over the Jacobi algebra. Also, the
V-filtration on Щ induces the V-filtration with VSif := pr(V П Щ) on
af.
Varchenko and Chmuto v [ChV] considered the tangent cone to the /U.-constant
stratum. It is estimated by the space
V\O/Jf) := {g 6 O/Jf | g ¦ Уа% С Va+1Qf}.
A2.18)
In fact, they took not only the У-filtration into account, but also some weight
filtration. Some remarks on that will come in section 12.2. Here we will see
that the space Vx(O/Jf) fits very well to the canonical complex structure on
Lemma 12.5 The space VX{O/ Jf) С O/Jf is canonically isomorphic to the
Zariski tangent space in TqM of the /л-constant stratum (S^, 0) with the canon-
canonical complex structure from A2.6).
Proof. Choose (?/., y\) an^ "l. ¦ • ¦ < vij. as in section 11.1. One arrives at the
following identity in Я/ for X 6 TM,o for example through A1.35), A1.33),
A1.49)
i )l,«o •["*]• A2Л9)
Because of Vuf = @a.>a C[u?], the space V\O/Jf) is isomorphic (viaa0)
to
{X|,_o еГ„М|Хе TM>0, X(cf/) \t=0 = 0 for aj-l- a,- < 0}. A2.20)

•г'
224
^-constant stratum
This is the Zariski tangent space of EM,0) with its canonical complex
structure. D
Remarks 12.6 (a) Choose (U,, y\) as in section 11.1. The functions r,- := cu
are flat coordinates on M with <5,- = -^ (Theorem 11.9 (b)). The д-constant
stratum is contained in the flat subspace of M
{t € M | Xj(t) = OfOTdj -1-Cti < 0}.
A2.21)
In general this is only a coarse estimate. But for a quasihomogeneous singularity
/ this flat subspace is the д-constant stratum. In that case E\,-o = 0, and E =
X^O+ai ~ctj)Tj8j by A1.57). All coefficients c\p are weighted homogeneous
with weights or,- + p — aj because of A1.58). Therefore then
- 1 - or, < 0) = (ту I ay - 1 - a, < 0) С
>M,0-
It coincides also with the ideal (tj | a,- — 1 — ori < 0). Varchenko [Va4] was
the first person to see that this last ideal gives the д-constant stratum.
(b) In the curve case (n = 1) the д-constant stratum is smooth ([Wah],
see also [Matt]). Unfortunately, it is not clear whether the canonical complex
structure is then the reduced structure. Results on the Zariski tangent space in
[ChV] indicate that this should be the case for (generic?) curve singularities
with nondegenerate Newton boundary.
(c) For л > 2 the д-constant stratum can have singularities [Lu][Stv]. The
construction in [Lu] is elegant, but it makes use of a part of the Zariski con-
conjecture, that the multiplicity should be constant along the д-constant stratum.
At the time of [Lu] a proof of it was announced which later turned out to be
wrong. So [Lu] and an also elegant generalization in [VaS] concern the sets
S/j, П {multiplicity constant}.
In [Stv] it is proved for one example of [Lu] that this (singular) subset is
at least a component of 5M, if it is not the whole space 5M. Results in [VaS]
indicate that this subset (and probably 5M itself) can have arbitrary singularities
(but the Milnor numbers in examples in [VaS] become astronomically high).
(d) One may also expect that the canonical complex structure of 5M can be
nonreduced. But examples are not known. Computing this canonical structure
for higher Milnor numbers is difficult.
12.2 Period map and infinitesimal Torelli
Let / be a singularity as in section 12.1 with semiuniversal unfolding F, base
space M, and ^-constant stratum 5Д С М. We argued in Remark 12.3 that
12.2 Period map and infinitesimal Torelli
225
the Brieskorn lattices H^t) for all t 6 5K can be considered as sublattices of
the same lattice V>~i. Also the Hodge nitrations F'(t) can be considered as
filtrations on the same space Я°°.
By Varchenko's result (Theorem 10.32, [Va3]) that the spectral pairs are
constant within the jx-constant stratum, these Hodge filtrations are contained
in the same classifying space Dpmhs for polarized mixed Hodge structures.
By definition, the classifying space DBL from A0.94) for Brieskorn lat-
lattices is a fibre bundle over DPMHS, with projection prPMHS : DBL -» DPMHS.
Therefore we obtain two period maps
Ф :
DBL, г н> Яо'@, A2.22)
DPMHS, t h+ F'(t). A2.23)
There is also a global period map. In section 13.3 we will discuss the space
MM := {singularities in one д-homotopy class}/right equivalence A2.24)
and prove that it has a natural structure of a complex space. The group G% =
Aut(#g°, h, S) acts on У>-1, respecting the pairing Kj and the actions of z and
Э. It acts on DPmhs and DBL properly discontinuously (Theorem 10.34).
The quotient DBL/G% is the moduli space for Brieskorn lattices, or more
precisely, for tuples (Я|°, h, S, V>-l, Hq) up to isomorphism. One obtains a
global period map
M, -* DBL/GZ. A2.25)
In [He2] the following global Torelli type conjecture was formulated.
Conjecture 12.7 The period map
fi-homotopy class of singularities.
is injective for any
It was proved in [Hel][He2] for all unimodal and most of the bimodal singu-
singularities. The only exceptions were some subseries of the 8 bimodal series where
the period map could not be determined precisely enough to see whether it is
injective. In [He3] it was proved for the semiquasihomogeneous singularities
with weights (i, 5,5,3) (with dim 5K = 5) andfor аЁ semiquasihomogeneous
singularities with weights (j-,..., ^-) such that gcd (at, aj) = 1 for all i ф j
(with dim 5K arbitrarily high). At to the present time no counterexamples are
known.
In section 13.3 we will see that the space ЛЛц, is locally the quotient of a
д-constant stratum by a finite group. The period map Ф : 5M -*¦ DBl is a local
lift of the global period map. M. Saito did not have the classifying space DBL,

'ill
'¦О
226
jx-constant stratum
but nevertheless he considered essentially the period map Ф, as a map into a
manifold which is a classifying space for subspaces of V>~[/V"~l with the
correct spectral numbers.
In [SM3,2.10] he proved that itis injective if S^ is smooth. In [SM4, Theorem
3.3] he used this to show that for any 5M it is finite-to-one if one chooses 5М
sufficiently small. He did not use the construction in section 11.1. With that
and with the precise knowledge of coordinates on Dbl we obtain the following
infinitesimal Torelli type result.
Theorem 12.8 The period map Ф : 5M -*¦ Dbl is an embedding o/5M with its
reduced complex structure into Dbl if SM is chosen small enough.
Proof: Choose an opposite filtration U, for the singularity / and elementary
sections si € Ca' as in section 11.1. We claim that any lattice К € DBl in a
neighbourhood of the Brieskorn lattice Щ' e Dbl has a unique element of the
type
vf =
A2.26)
for / = 1,..., ix. One constructs it in two steps, first the principal part, then
the rest. The filtration U, is also opposite to the Hodge filtration prpMHs(K) €
Dpmhs- So one obtains spaces GaK as in A1.5). They satisfy the analoga of
A1.6)—A1.10) and also
a г
A2.27)
p>0
and vice versa. There is a unique elementary section sf € G^ with sf — Si €
= Si +
p: P>\.aj-P=at
A2.28)
The element vf is built from the sections sf by the same procedure as in A1.24)
and the proof of Theorem 7.16. Uniqueness is clear from the construction.
The sections vf generate К as а С{{Зг}}- module and as a C{z}-module.
The lattice К and the coefficients у>р{К) determine one another uniquely,
i
The period map
Dbl sends t e 5M to the lattice К е DBl
p p
with y[F\K) = c\f(t). Now the injectivity of the period map follows simply
from the facts, that the coefficients c\- for у = 1,...,/x are flat coordinates
on M (Theorem 11.9 (b)) and that 5M С {r | c^J = 0foray - 1 - ax < 0}
(Theorem 12.2).
12.2 Period map and infinitesimal Torelli
227
To see that the period map is an embedding is much more difficult. We will
show the existence of an index set / С {(/, ;, pY\ otj - p - a, > 0, p >
0} and of a submanifold M С M with 5M_C M such that the coefficients
\v! di M d h h ffii {jp |
} a
c\v!
nd of a submanifold С M_
|A, j, 1) € /} serve as coordinates on M and that the coefficients
{c\ |A, j, 1) } yjp
(i, j, p) € /} serve locally as coordinates on Dbl- That is obviously sufficient.
The existence of an index set / such that {y-f | (i, j, p) e 1} serve locally
as coordinates on DBl follows from the construction in [He4]. It decomposes
into the index sets h - {(«. h P) e 7 I aj ~ P ~ a'" = °J and h = l ~h- The
set h corresponds to coordinates for the fibres of prpMHS '¦ Dbl -*¦ DPMhs-
It can be chosen explicitly as
h = {(i, j, 1) I' < Д + 1 -
aJ ~ l ~ «'¦
°Ь
A2.29)
compare [He4, Theorem 5.6] and A1.51), A1.52), A1.55), and Remark 11.10
(b). It especially contains all triples (I, j, 1) with aj — 1 — a,- > 0.
The set I\ corresponds to coordinates for the base DPmhs- Unfortunately, an
explicit choice is difficult and not provided in [He4]. We now have to construe t it.
Let us first forget about the si and yfp and find a basis af of Я°° and
coefficients f}^ with ay — p - a, = 0, which are better suited.
021 h i b lp
p'4
and
Because of the proof of Lemma 10.21 there exist subspaces l
with all the properties in Lemma 10.18 except A0.51)+A0.52) and which give
the opposite filtration Um by the analogon of A0.66). One can choose bases
of all the primitive subspaces (Iq'4)x such that they together with their images
under powers of N form a basis a\,..., a° with the following properties: set
a°+1 := 0; there exists a map v : {1,..., ll} ->¦ {1,..., /x, д + 1} and an
involution к : {1,..., /л] -*¦ {1,..., д} with
N af = a%y
a? €
П
A2.30)
A2.31)
A2.32)
here FJ denotes the Hodge filtration for /. The basis a\,..., a° is a basis of
Jordan blocks which are dual or selfdual with respect to the polarizing form S
and which fit to the splittings and strict morphisms in Lemma 10.18. Note that
N is an infinitesimal isometry of S.
For any Hodge filtration F' € DPMHS sufficiently close to Fo*, there exists
a unique basis a\,..., a^ of #°° with
a,-
a. eF[n-a']n U[n-ai]
a? € U[n-a,]-\-
A2.33)
A2.34)

228
It has the form
fi-constant stratum
: p>0,ocj—p—ai=0
A2.35)
One checks as in the proof of Lemma 10.21 that the a\,..., aM satisfy the same
relations A2.30) and A2.31) with respect to S and N as the a°,..., a°. This
gives relations between the coefficients p\p. The relations from N say:
(a) each coefficient f}\f with a? e ImN is zero or equal to a coefficient (iff
with а^ <? ImN,
OS) such a coefficient #[f> with a? ? ImN is zero if a°k e kerN and a° ?
кегЛР1 for some m.
One eliminates with (or) all coefficients fi\f with a, e ImN. With respect to S,
observe that N is an infinitesimal isometry. One checks easily that the relations
from S satisfy:
(y) All the nonvanishing different relations have one or two coefficients as
linear terms and otherwise quadratic terms; in one relation the upper indices
or sums of two upper indices are all equal; any coefficient turns up as linear
term in at most one relation.
In Definition 10.16 of a polarized mixed Hodge structure, the conditions (i)
and (iv) (/3) are open conditions. The others are satisfied for a filtration which
is defined by a basis ab ..., aM as in A2.35) whose coefficients p\f satisfy the
relations from the above N and S. In view of (а)-(к), certain of the coefficients
$p form local coordinates for DPMHS. We are especially interested in the
behaviour of the coefficients /?{'•':
(S) certain of them might be zero because of (/3),
(e) some others might be zero or linearly related only to one another because
of(K),
(?) all others can be chosen as coordinates.
Now we return to the coefficients у^\ The relations in (S) and (e) induce linear
relations between the у.Ф for or, - 1 - a, = 0. The index set /, can be chosen
to contain the indices of all y,y with a,- — 1 — ct\ — 0 except those which can
be eliminated by these relations.
The corresponding linear relations between the c\l? with Uj — 1 — a\ =0
define the submanifold M С [t e M \ c^J = 0 for a, - 1 -ai < 0}. Obviously
SuCM. П
12.2 Period map and infinitesimal Torelli
229
Remarks 12.9 (a) It is not clear whether the period map Ф : 5M -> DBL is an
embedding of 5M with its canonical complex structure into Dbl- The reason is
that it is not clear whether SM with its canonical complex structure is embedded
in the flat submanifold M which was constructed in the proof. One does not
know whether the corresponding relations between the c^j with a, — 1 - a, =0
from (E) and (e) in the proof are contained in the ideal in Theorem 12.2.
(b) It might be that such extra relations do not turn up or that they are always
contained in the ideal in Theorem 12.2. It might also be possible to define in a
natural way an a priori bigger ideal which takes such relations into account. But
one would need more elaborate choices and independence of choices would be
a much bigger problem.
An infinitesimal refinement of Vl(O/Jf) which takes into account relations
from the Jordan blocks and the weight filtration (i.e. type (/3) and (S) in the
proof) has been discussed in [ChV].
(c) The sections vf in A2.26) are very similar to the sections t>,- in chapter 11.
One has to replace M by DBL. The sections vf glue to generators of a free
CPi x Dbl -module over P1 x DBt, which is the sheaf of sections of aflat vector
bundle over C* x DBL. The restrictions to (Д x [K], @, K)) generate K. Along
{oo} xDbl the sheaf has a logarithmic pole, and the restrictions to {oo} x Dbl of
the sections vf are flat with respect to the residual connection along [oo}xDBL.

J !
Chapter 13
Moduli spaces for singularities
The choice (?/., yO in Theorem 11.1 induces isomorphisms between spaces
which are not canonically isomorphic. Section 13.1 fixes this and the compatibil-
compatibilities of these isomorphisms with respect to coordinate changes and /л-constant
deformations.
This is used in section 13.2 for a discussion of symmetries of singularities
and in section 13.3 for the (up to now) best application of the construction of
Frobenius manifolds in singularity theory: the existence of global moduli spaces
for singularities. This really comes from an interplay between polarized mixed
Hodge structures and Frobenius manifolds, with the construction in section 11.1
as the link.
13.1 Compatibilities
A choice (?/., y{) as in Theorem 11.1 for a singularity / does not only in-
induce the metric of a Frobenius manifold. It also induces isomorphisms be-
between several /u-dimensional spaces which are not canonically isomorphic
(Lemma 13.3). These spaces and isomorphisms behave naturally with respect to
/Lt-constant families (Lemma 13.4) and with respect to isomorphisms of singu-
singularities (Lemma 13.6). A part of this is elementary, a part is a direct consequence
of the construction in section 11.1. АЁ of it fits to the expectations. But to use
it in sections 13.2 and 13.3 we need precise notations and statements. First,
in Theorem 13.1 and Theorem 13.2 some results from part 1 are summarized
which will be needed in sections 13.2 and 13.3.
Theorem 13.1 Let {{M, t), o, e) be the germ of a massive F-manifold with
multiplication о and unit field e. The group Aut((M, t), o, e) of automorphisms
of the germ of the F-manifold is finite. If E is an Euler field of the F-manifold
(i.e. Lie?(o) = 1 • o) such that the endomorphism TtM -» TtM, Ik?oI,
is nilpotent, then Aut((M, t), о, е) — Aut((M, t), о, е, Е).
230
13.1 Compatibilities
231
Proof. The finiteness is Theorem 4.14. An Euler field is mapped to an Euler
field by an automorphism of an F-manifold. If the germ (M, t) has an Euler
field, then there is a unique Euler field whose action on T,M is nilpotent.
This follows from Theorem 3.3. Any automorphism of the germ maps this
Euler field to itself. D
Theorem 13.2 Let f(x0, ...,xn) and f(x0,..., xn) be two isolated hyper-
surface singularities with semiuniversal unfoldings F and F and germsofbase
spaces (M, 0) and (M, 0) as in section 10.1. The germs (M, 0) and (M, 0) are
germs of massive F-manifolds.
(a) A germ of a biholomorphic map <p : (Cn+1, 0) -*¦ (C+1,0) (i.e. a co-
coordinate change) with f — f о <р can be extended to an isomorphism of the
unfoldings, that is, a pair of isomorphisms (Ф, q>M) such that the diagram
commutes
(C+1 x M, 0)
[pru
(M,0)
(C+1 хМ,0)
A3.1)
4>u
(M, 0),
and
Ф|(С"+1х@},0) — '
F = F оФ
A3.2)
A3.3)
hold. The map <pM is an isomorphism of germs of F-manifolds. The map Ф is
not unique, but <рм is uniquely determined byjp.
(b) For any isomorphism <pM ¦ (M, 0) -» (M, 0) of the germs of F-manifolds
there exists а Ф such that the pair (Ф, ipM) is an isomorphism of the unfoldings,
that is, it satisfies A3.1) and A3.3) (Ф is not at all unique).
Proof, (a) The existence of an isomorphism (Ф, срм) of the unfoldings which
extends cp is classic, it is essentially part of the definition of a semiuniversal
unfolding (e.g. [Was], [AGV1, ch. 8]). The base map <pM is an isomorphism
of germs of F-manifolds, because the Kodaira-Spencer maps, which define the
multiplications, are compatible with the pair (Ф, срм) (E.10) and Remark 5.2
(v)). It is unique because the tangent map at 0 is unique and the germs of
F-manifolds are so rigid by the above Theorem 13.1 (Theorem 5.4).
(b) Theorem 5.6 (b)). ?

232
Moduli spaces for singularities
Now we turn to the construction in section 11.1. Let f(xo,.. ¦, х„) with
n > 1 be an isolated hypersurface singularity as in chapters 10 and 11 and
F : (C"+1 x M, 0) -» (C, 0) a semiuniversal unfolding, H" the cohomology
bundle over A x M — V (for a representative of F), 7^@) the extension from
Lemma 10.2 to A x M of its sheaf of holomorphic sections, Щ' the Brieskorn
lattice, H°° the space of global flat multivalued sections on Я"|(Л х {0}, 0),
and F* Steenbrink's Hodge filtration on it.
Consider the four д-dimensional spaces TqM, O/J/, Qj, and H°°. Here
O/J/ is the Jacobi algebra and (cf. A0.87))
Й, =
л
The reduced Kodaira-Spencer map ao : T0M -*¦ O/Jf (cf. A0.1)) is a canon-
canonical isomorphism. The space uf is a free 0/-//-module of rank 1; generators
are the classes in Й/ of volume forms. One has to choose such a class of a
volume form to obtain an isomorphism O/J/ = ?1/. There are no obvious
natural isomorphisms between H°° and the other three spaces.
Lemma 13.3 A choice (U,, y\) as in Theorem 11.1 induces the isomorphisms
., у\\ in the following diagram
TcsM
O/Jf
4
A3.4)
The map i/f| comes from the splitting Я00 = 0p Fp П Up, the map 1/^4 is the
O/Jf-module isomorphism with V4(l) = [v?] (cf A1.16) and A1.17)), the
map 1/^3 is the composition of the projection Щ -*¦ ?1/ with an embedding
A3.5)
which will be explained in the proof. Denote by
ToM
A3.6)
the induced isomorphism. The maps ijf\, 1/^2, and 1/^3 depend only on U., the
maps iff4 and 1/^5 depend also linearly on y\, that means, for с 6 С* one has
A3.7)
Proof. For the definition of i/r2[U.] one has to be well aware of the construction
in section 11.1 up to A1.17). The map 1/^2 is composed of the isomorphisms
1/r
f1
Gr"pH°
UPH°°, A1.5), A1.13), A1.4), and the restriction of
13.1 Compatibilities
233
global sections in ft@) to sections in the Brieskorn lattice Щ. Properties of fa
will be discussed in the proof of Lemma 13.4
The dependence on yi of the isomorphisms follows from i/r3(yi) = [u°]. ?
Lemma 13.4 Fix a choice (U., y\) as in Theorem 11.1. Consider two points h
and h in the ^-constant stratum 5M С М. The spaces Я00 of the corresponding
singularities are canonically isomorphic and will be identified (cf. Remark
12.3).
By Theorem 11.1, the opposite filtration U. (without y\) induces aflat struc-
structure on M. This yields an isomorphism
<y[U.,tut2\:ThM^ThM.
A3.8)
One has for the singularities for t\ and ti isomorphisms as in A3.6). Denote
them by ^sKU., y\), t\] and ^sKU., yi), tj\ They are compatible with A3.8):
A3.9)
a[U., tu ь] о fs[{U., yx), ty] = Ши., П), hi
Proof. One has to review the definition of 1/Г5. The isomorphism 1/^ ' and the
spaces G" of elementary sections in A1.5) depend on the Hodge filtration,
which may vary for the singularities in 5M. But the composition of i/лj~ ',A1.5),
and A1.13) is independent of the Hodge filtration. With A1.4) one arrives at
the same isomorphism
• vi A3.10)
forallsingularitiesinS^.Onenowhastoseethatforanyf 6 S^ the isomorphism
фС- v,- -> T,M, Vi^Si\T,M > A3.11)
coincides with the composition of the restriction of u,- to the Brieskorn lattice
HqQ) and the isomorphisms 1/r^1 and ap1 for the singularity corresponding to
t e Sp. This follows with Lemma 11.5 (b). , D
Remarks 13.5 (a) The isomorphism i/^i depends on the Hodge filtration of a
singularity. Therefore, in general, one does not have a compatibility of the iso-
isomorphisms 1/Г5 о i/^! [tj] : Я°° -*¦ Tu M, i = 1, 2, for two parameters t\, ti € SM
with A3.8).
(b) In the diagram A3.4) of isomorphisms the semiuniversal unfolding is
needed only for the Kodaira-Spencer isomorphism ao. In the definition of ifo
one can replace A1.13) and A1.4) by the corresponding restrictions to the slice
P' x {0} С Р' x M, or one can define the section vf directly as the unique

234
Moduli spaces for singularities
1
iic
x\
¦<
tl
га
section in Щ with principal part s,- and higher elementary parts as in formula
A1.24).
Lemma 13.6 As well as f consider a second isolated hypersurface singularity
f = /(*o. ••-,*„) which is right equivalent to f. Choose a semiuniversal
unfolding F with base (M, 0)for f and write all the other associated objects
for f with a tilde. Fix a coordinate change cp : (C"+1, 0) -*¦ (C+1, 0) with
f = f°<P-
(a) It induces isomorphisms between the corresponding objects for f and f,
which respect all the canonical additional structures,
<pM : ((M, 0), o, e, E) -» ((M, 0), o, 7, E),
4>Jac : (O/Jf, mult., [/]) -* (O/Jj, mult., [/]),
cpcro : (П/, Jf) -* (П?> Jy),
Van : {Щ, z, d~\ Kt) -* (Щ, z, Ь~\ Kj),
<pcoh : (й|°, h, S, F') -* (H™, h, S, F'),
(cf. section 10.6 for the residue pairing Jf, the polarizing form S and the form
Kf).
(b) Choose (?/., yi) as in Theorem 11.1 and set U. := <pcoh(U.) in H°°,
Y\ := (Gr^cpcohXyO. The diagram A3.4) for (/, ([/., yx)) and the diagram
A3.4) for (/, (U., y\)) are compatible with the above isomorphisms and with
the induced isomorphisms.
The same holds for the embeddings fo and \jf2 from A3.5). The map <рм is
an isomorphism of Frobenius manifolds with respect to the metrics on M and
M induced by(U.,y\) and ([/., y\).
Proof. This holds, because all the considered objects and structures and also
the construction in section 11.1 are essentially coordinate independent.
Theorem 13.2 provides the uniqueness of cpM. For the other isomorphisms
one does not even need to consider the semiuniversal unfoldings because of
Remark 13.5 (b). A nontrivial point is to only formulate one or several nat-
natural definitions for the isomorphisms <рм, ••¦', <pCoh and to check that all are
compatible. See Remark 13.7. ?
Remark 13.7 For example, <рв„ can be defined by pulling back differential
forms with (ср~1)*. It can also be recovered from <рс„&: By sections 7.1 and
10.6, the tuple (H°°, h, S) determines the structure (V>, z, d~\ Kf). The
13.2 Symmetries of singularities
235
isomorphism cpcoh induces an isomorphism from (V> ', z, dz ', Kf) to the cor-
corresponding structure for /. It maps Щ to H^.
Another example: 0/7/ is naturally embedded in End^/_ adjoint(Q,f, Jf) by
the action on ?2/. Knowing this embedding for / and /, one can recover cpJac
from cpGm.
13.2 Symmetries of singularities
The following is motivated by the results of Slodowy [SI] and Wall [Wall] and
an extension of them. Remarks on the relations will be made below.
Let / : (C+1, 0) -» (C, 0) be an isolated hypersurface singularity with
n > 1, П := {<p : (C"+1,0) -> (C+1,0)biholomorphic} the group of all
coordinate changes, and TZf = [tp e U \ f о q> — f) the group of symmetries
of/.
The group of symmetries acts on all the objects associated to the singularity.
The interrelations presented in section 13.1 result from heavy machinery and
profound facts, but now they make it easy to analyze the actions of the symmetry
group.
The group TZf is oo-dimensional, but the group of fc-jets jkR-f is an algebraic
group for any к > 1. Let
A3.12)
be the finite group of components of j\R.}. The following is classical.
Lemma 13.8 The kernel kev(jkTZf -> jiTZf) is unipotent. The groups jkTZf
and j\VJ have the same number of components.
Proof. The group jkTlf acts faithfully on 0c«+i,o/m*+\ where m is the maxi-
maximal ideal. An element of the kernel acts trivially on m/m2 and thus on m' /m'+1
for any / < k. It is unipotent. The kernel is unipotent and thus connected. ?
We use the following abbreviations for the groups of automorphisms of
different objects associated to the singularity.
AutM = Aut((M, 0), о, е, Е), A3.13)
Autyac = Aut(C/7/, mult., [/]) = AutGbM, o, ?|0), A3.14)
Gz = Aut(H^,h,S), A3.15)
StabGz(F") = Aut(H^, h, S, F'), A3.16)
StabCz(flSO = Aut(ff|°, h, S, V"-1, Hg). A3.17)

236
Moduli spaces for singularities
Here (M, 0) is the base of a semiuniversal unfolding, a germ of an F-manifold
with Euler field, O/Jf is the Jacobi algebra, Я|° the lattice in the space Я°° of
global flat multivalued sections in the cohomology bundle over a punctured disc,
h its monodromy, S the polarizing form from section 10.6, Щ the Brieskorn
lattice, and F* Steenbrink's Hodge filtration on H°°.
One can see that Gz is isomorphic to the automorphism group of the Milnor
lattice with Seifertform[He4]. The group Gz acts on(V>~l,z, d~l, Kf)(com-
pare sections 7.1 and 10.6). It acts properly discontinuously on the classifying
spaces DPmhs and DBl (Theorems 10.22, 10.34).
Therefore the groups Stabcz(#o') С Stabcz(F*) are finite. The group Autjv/
is also finite (Theorem 13.1). Because of Theorem 13.2 and Lemma 13.6 there
are canonical homomorphisms from the group Kf of symmetries of / to the
groups in A3.13)—A3.17). Denote
deto : Uf -»¦ C*, <pv+ det (—\ @).
A3.18)
Because of the splitting lemma one can transform any singularity by a coordinate
change to the form in Theorem 13.9.
Theorem 13.9 Suppose that f = g(x0, ...,хт)+х„+1-\ Yx\withg e m3
for some m < n.
(a) The canonical homomorphismsfromiV to the groups in A3.13)—A3.17)
factor through Rf.
(b) If n = m then the map Rf -*¦ Autji/ is an isomorphism. Ifn>m then
Rf = Rgy. %2, the map Rf —> Aut*f is two-to-one, and the kernel is generated
by (x0,..., х„_ i, х„) h* (xo,..., xn-1, -xn). In both cases (n > m), the map
Rf —> Autyac is induced by the embedding of Aut« in AvA{TqM, o, ?|o) =
AutJac.
(c) Ifn = m then Rf = j{Rt and any maximal reductive subgroup of' j^RJ
is isomorphic to it. Ifn > m then any maximal reductive subgroup of j]Ji.f is
isomorphic to Rg x O(n — m, C).
(d) The map deto factors through Rf. The action of Rf on H°° is obtained
from the action on TqM by twist with det^. More precisely, there is an isomor-
isomorphism f6 : H°° -» T0M such that for all <pe1lf
<Pcoh = V*6 ° det0 («0
A3.19)
Here <pcoh € Gz and <рм ? Aut^ are the induced isomorphisms.
(e) The map Rf —>• Stabcz(HQ) is injective, so Rf acts faithfully on the
Milnor lattice.
13.2 Symmetries of singularities
237
Proof, (a) The group Aut^ is embedded in Aut(T0M, o, ?|0) = Autyac, be-
because it is finite. The actions of IZ? on ToM and on O/Jf are isomorphic via
the Kodaira-Spencer isomorphism ao. The action on O/J/ depends only on
sufficiently large fc-jets and depends continuously on them, the action on TqM
is finite. Therefore both factor through Rf.
The same argument works for H°° and fi/ with a special choice of U,: The
opposite filtration U^ from Deligne's /?•? on H°° (cf. A0.66))
[/(°>=
A3.20)
is invariant under <pCOh for any <p e Л?, because the 1рл are invariant by their
definition (cf. A0.50)). It induces an isomorphism ^3 о ф{: Н°° —»• fi/ by
Lemma 13.3. The actions of Tlf on Й/ and on H°° are compatible with it by
Lemma 13.6. But StabGz(//o') is finite and the action on fi/ depends continu-
continuously on sufficiently high &-jets. So both factor through Rf.
(b) and (c) First, suppose n=m. Then /em3 and there is a natural surjec-
tive map m/7/ —>• m/m2. The group action of j\Rf on m/m2 is faithful. It
factors through Rf by (a). Thus j\Uf ~ Rf and Rf acts faithfully on O/Jf.
Therefore Rf -» Aut« is injective. It is surjective because of Theorem 13.2
(b). Because the kernel ksr(jkTZf -> jiTZf) is unipotent (Lemma 13.8), any
reductive subgroup of jkTZf is mapped injectively to Rf.
Now, suppose n > m. A symmetry <p e 72/ acts on Cc»+',o by h ну h о <р~х.
It leaves the Jacobian ideal invariant. The ideal m2 + 7/ = m2+(xm+\,... ,xn)
is invariant under the action of 72/. There is a natural homomorphism
]{
Aut
m
+ (xm+
x Aut I
...,xn)J \
m2
A3.21)
The kernel is unipotent, the image is isomorphic to Rg x O(n — m, C), because
j2f = x^+l + ¦ ¦ ¦ + *2. Therefore Rf = Rg x Z2. The other statements are
also now clear.
(d) Choose any generator y\ of the 1-dimensional space Gr^_a|]W°°WB|. By
Lemma 13.3 (U®\ y\) induces a class of a volume form [d°] e Я/ and an
isomorphism 1/4 : O/Jf -» fi/, [1] н* [v°], of C/7/-modules. Also fiiyi) =
[«?]•
The actions on Gr17 Я°° and fi/ of a symmetry <p e 72.-^ are compatible with
1Д3 and have yi and [d°] as eigenvectors. The eigenvalue is det^'(#>), because
[v\\ is represented by a volume form. By Lemma 13.6 the actions <pjac and <paf
are compatible with 1^4 with the twist det^' (<p\
° <PJac = <Pa,
A3.22)

238
Moduli spaces for singularities
Consider A3.4) and define
:= a
'
'
о fa о fa. The compatibilities of
the actions of <p with a0 and fa о fa show that the twist extends to A3.19). The
map deto factors automatically through Rf.
(e) The kernel ker(/J/ -> Gz) is contained in the kernel of deto because of
the action on С ¦ yx in (d). Then by A3.19) it is the intersection of the kernels
of deto and Rf -> h.\AM. But this is trivial. D
Remarks 13.10 (a) Slodowy [SI] considered a real singularity with a given
compact group of symmetries and showed that one can construct a semiuniversal
unfolding on which the group acts. He also showed that such a group is finite if
/em3. Compact corresponds to reductive in the complex case. So his results
are close to Theorem 13.9 (b) and (c). But he did not have the uniqueness
of the action on the basis (M, 0), which follows from the rigidity of massive
F-manifolds (Theorem 13.1), and he did not consider Rf and AaX.M-
(b) Part (d) in Theorem 13.9 is due to Wall [Wall], also the way in which
this is proved. But he did not have the up to a scalar canonical isomorphism
fa : H°° -> TqM, which comes from the construction in section 11.1 and from
Deligne's /"•?.
(c) In [SI] and [Wall] one starts with a given (compact or) finite subgroup
of TZf. The group Rf and its liftability to a subgroup of lZf are not considered.
But this liftability can be shown easily, following arguments in [WaI2] (see also
[Muel]):
It is known that the group Rf can be lifted to a subgroup Rf in the algebraic
group jkTZf for any k. Let к > /м +1. Within the fc-jets one carries out the usual
averaging procedure to find coordinates on which the group Rf acts linearly
and obtains а к -jet jkip of a coordinate change such that jkcp~l о Rf о jk<p acts
linearly on O/mk+l and respects jkf о jkip. The last group lifts to a group of
linear coordinate changes and respects jkf о jk<p, considered as a polynomial.
This polynomial is right equivalent to /, because / is д + 1-determined.
In the case of a quasihomogeneous singularity / one can calculate the group
Rf using the following characterization.
Theorem 13.11 Let f e C[x0,..., xn] be a quasihomogeneous isolated sin-
singularity with weights wo,...,wn 6 @, |] П Q arid degree 1. Suppose that
wo < • • • < wn_] < I (then /em3 if and only if wn < \). Let Gw be the
algebraic group of quasihomogeneous coordinate changes, that means, those
which respect C[*o, • • •, xn] and the grading by the weights w0,..., wn on it.
Then
Rf = StiibGJf).
A3.23)
13.2 Symmetries of singularities
239
Proof. The group StabG.,(/)is nnite ЬУ [GrHP, Proposition 2.7] (the proof
is similar to that in [SI, D.6)]). In [GrHP, Theorem 2.1] it is proved that any
symmetry cp e TZf has weighted degree > 0, that means, the г-th component
cpi e О does not contain monomials of weighted degree < w,. The degree
0 part of any symmetry is an element of StabG,,,(/).
One can rewrite Lemma 13.8 for weighted jets and for StabGl,.(/) instead
of }{RJ. But the groups of jets and of weighted jets of symmetries are con-
contained in one another for suitable high degrees. They have the same number of
components. This shows A3.23). D
Conjecture 13.12 The map Rf -у StabGz(#o) is an isomorphism for any iso-
isolated hypersurface singularity f(x0, .. ¦, xn) with n > 1 and multiplicity 2.
Theorem 13.13 Suppose that the map Rf ->¦ StabGz(Wo') is an isomorphism
for one singularity f(x0, ...,xn) with n > 1 and multiplicity 2.
Then the corresponding map is an isomorphism for any singularity in the
(sufficiently small) ix-constant stratum 5M of f.
Proof. We may choose a (sufficiently small) representative F of the semiuni-
semiuniversal unfolding, a representative in Hf for each element of Rf, and extensions
of these representatives to automorphisms of the unfolding F in the sense of
Theorem 13.2 (a).
Consider the period map S^ -> DBL, t м>- Н?Ц), from section 12.2. If SM is
sufficiently small, then StabGz(#?(O) С StabGz(Ho'@))foranyr е S^ because
Gz acts properly discontinuously on DBL (Theorem 10.34).
By assumption, an element cpcoh 6 StabGz(Wo (О) С StabGz(tfov(O)) = Rf
is induced by an element of TiJ. It induces an automorphism cpM of M. By the
compatibility of the morphisms in Lemma 13.6, <pco/,(ffo'@) = Щ(<Рм^У).
But the period map is injective (Theorem 12.8). Therefore <pM(t) = t.
The automorphism of the unfolding restricts to a symmetry of the singular-
singularity F,. This symmetry induces cpcoh. Therefore RFl -у StabGz(Wo'@) is an
isomorphism. ^
Remarks 13.14 (a) Conjecture 13.12 complements the global Torelli type
conjecture 12.7. Together with the infinitesimal Torelli type theorem 12.8 and
Theoreml3.15belowitwouldsaythatlocallytheperiodmapyHM -*¦ DBL/GZ
is an embedding. The only obstructions to Conjecture 12.7 would be possible
intersections of the images of disjoint pieces of Мц.
(b) The conjecture is true for the weighted homogeneous singularities with
weights (^,..., ^) where gcd(ai,aj) = 1 for i ф j.

X
I "О
!-<
л/У
О
от
240
Moduli spaces for singularities
For these singularities all eigenspaces of the monodromy are 1-dimensional
and the orders of their eigenvalues can be put into certain chains with one biggest
order. This is used in [He3, Proposition 6.3] to show that StabcT(#g') = G% =
{±hk | к e Z). This is isomorphic to StabCi/,(/) if an = 2.
(c) The same reasoning as in (b) applies to the Ад., D2k+\, Ek, and to 22 of
the 28 quasihomogeneous unimodal and bimodal exceptional singularities, but
not to D2k, Zn, Qii, Ul2, Zuj, 2,6, Ul6.
Starting with a Coxeter-Dynkin diagram, I checked Conjecture 13.12 for
Q\2, which has some 2-dimensional eigenspaces and Stabc,,(#o') = ^z =
{±hk | A e Z} x Z2 (= Rf for n > 3, = StabC]i,(/) for и = 3).
13.3 Global moduli spaces for singularities
In this section we present our best application of the construction of Frobenius
manifolds for hypersurface singularities: Theorem 13.15 gives a global moduli
space for the singularities in one /x-homotopy class.
The three main ingredients in the proof are the F-manifold structure on a
base M of a semiuniversal unfolding of a singularity, the construction of fiat
structures on M by choice of opposite filtrations, and the period map 5Д —>
Dpmhs from a ц.-constant stratum to the classifying space for polarized mixed
Hodge structures. So it grows from an interplay between Frobenius manifolds
and polarized mixed Hodge structures with the GauB-Manin connection as the
intermediary.
The construction of moduli spaces for singularities starts with Mather's the-
theory of jets of singularities ([Mathl][Math2], cf. [BrL]). The Ar-jet of a function
f e О = 0<c»+i,o 1S tne Class jkf € O/mk+1; here m С О is the maximal
ideal.
The action of the group П = {<p : (C"+t, 0) ->¦ (O+I, 0) biholomorphic } of
coordinate changes on m2 С О pulls down to an action of the algebraic group
jkTZ of A:-jets of coordinate changes on m2/ini+1.
An isolated hypersurface singularity /em2 with Milnor number /x is /x +1-
determined, that means, any function g with yM+, g = j^+if is right equivalent
to /. Therefore for A > /x + 1, the set of all 7?.-orbits in m2 of singularities with
Milnor number /i is in one-to-one correspondence with the set of /^-orbits of
their A-jets in m2/m*r+l.
Fix/i and A: > /x + l.Thecodimensionof the orbit j1lK'/i/mm!/ni'+' fora
singularity/with/x(/) = /x is д — l. The union of all orbits with codimension
> ijl — 1 is an algebraic subvariety in m2/m*+l. The set {jkf | /x(/) = /x) is
Zariski open in it and thus a quasiaffine variety. It decomposes into algebraically
irreducible components and into (possibly bigger) topological components.
13.3 Global moduli spaces for singularities
241
Each topological component corresponds to a /x-homotopy class of singular-
singularities. The singularities / and g in m2 with /x(f) = fi(g) = /x are /x-homotopic,
f ~м g> if and only ^tnere is a family / 6 m2, t e [0, 1], with jx{f) = /x,
/o = /, /, = g, such that the coefficients of the power series f, depend contin-
continuously (or, equivalently, C00 or even piecewise real analytic) on t. A singularity
/ with /x(/) = /X is /i-homotopic to its A-jet by the proof of the finite determi-
nacy, and A-jets in one topological component of the set above (Л / | /x(/) = /i}
are obviously д-homotopic. Finally, the group j^TZ is connected and acts on
each topological component.
For a singularity /6 m2 with /x(f) = /x, denote by C(A, /) the topological
component of [jkf | /x(f) = /x] which contains /. We summarize: the map
{g e m2 | g ~M f)/n -> C(A, f)/jkU
A3.24)
is bijective, the algebraic group jkTZ acts on the quasiaffine variety C(A, /) С
m2/m*+1.
Theorem 13.15 Fix /x, к > /x + 1 and a singularity / e m2 С Сс»+' ,o with
(a) The quotient М„. of the map ж : С (А, /) -»• C(k, f)/jkU =: 74M is
an analytic geometric quotient. That means, the quotient topology is Hausdorff
and the sheaf
A3.25)
A3.26)
induces a reduced complex structure on
(b)Thegermat[jkf]is
ц, [jkf]) = OS*, 0)/Aut((M, 0), о, е, Е).
Here ((M, 0), о, е, Е) is the base space of a semiuniversal unfolding of f with
its structure as a germ of an F-manifold (with Euler field), (SM, 0) С (М, 0) is
the ^.-constant stratum.
(c) The canonical complex structure on the ix-constant strata from Theorem
12.4 induces a (possibly nonreduced) canonical complex structure on M^.
The proof will be given after Remark 13.19. Results of Gabrielov [Ga] and
Teissier [Tel] will allow semiuniversal unfoldings to be considered instead
of spaces of A:-jets. Then Theorem 13.17 shows that the quotient topology is
Hausdorff. Theorem 13.18 gives the reduced complex structure and A3.26).
Remark 13.16 (a) All orbits in C(k, /) have the same dimension and are
closed. But this by no means implies that the quotient is Hausdorff. For example,

a
242
Moduli spaces for singularities
the quotient of C2 - {0} with the action of C* ^ GL{2, C), x i->- (J^,),on
it is not Hausdorff, the punctured coordinate planes cannot be separated.
(b) One may ask whether M^ is even an algebraic geometric quotient. But
it is not at all clear how one could approach this question. In general, the group
jkTZ is not reductive and the variety C(k, /) not affine.
(c) I expect that M.^ with the canonical complex structure in Theorem 13.15
(c) is a coarse moduli space for an appropriate notion of д-constant deformations
over arbitrary bases. But this has still to be worked out and checked.
Theorem 13.17 Let f and f e 6>c«+> ,o be two isolated hypersurface singulari-
singularities with (representatives of) base spaces M and M of semiuniversal unfoldings
and with ^-constant strata S^C M and 5M С М. Suppose that there are two
sequences(tj)iej<iand(tj)jew with Ъ e S^.Ti e 5M, f,-—»¦ OandTj —»¦ Ofori —> oo,
such that for each i € N the singularities which correspond to ti and ti are right
equivalent.
Then f and f are right equivalent.
Proof. We may suppose n > 1, because for и =0 there are only the Ад-
singularities. Let F and F be (representatives of) semiuniversal unfoldings of /
and /over M and M. Denote by F,,. = F|(C"+1 x {f,}, (x(i), f,-)) the singularity
which corresponds to tt; here x(l) is its singular point and F,,((jt(l), r,)) = 0
by definition of 5M. Define Fj. analogously and choose a coordinate change
(pi : (Cn+\xm)-+ (Cn+1,3c<l'))with Ftj = Т^ощ.
There is no possibility of controlling directly the sequence (#>,),€n of coor-
coordinate changes and finding a limit coordinate change for / and /.
But we have induced sequences of isomorphisms on several related objects
which can be controlled when they are seen together and which give the desired
information.
First, the germ F : (C+1 x M, (x(i), tt)) ->¦ (Cn+1,0) is a semiuniversal
unfolding for Fti, the same holds for F and Fjt. By Theorem 13.2 (a) q>t induces
an isomorphism of germs of F-manifolds
We will show that a subsequence tends to a limit isomorphism
<Роо,м ¦¦ W, 0) -> (M, 0)
A3.27)
A3.28)
of germs of F-manifolds.With Theorem 13.2 (b) or with Scherk's result
([Sche2], cf. Remark 5.5 (iv)) one concludes that / and / are right equivalent.
13.3 Global moduli spaces for singularities
243
In order to control the sequence ((pi,M)ieN, we need the flat structures and
their construction on M and M. Let H°° denote as usual the space of global flat
multivalued sections in the cohomology bundle over a punctured disc for /. It
is canonically isomorphic to the corresponding spaces for the singularities in
the //.-constant stratum 5M and will be identified with them. It is equipped with
the lattice #^°, the monodromy h, the polarizing form S (cf. section 10.6) and
for each t e 5M with Steenbrink's Hodge filtration F'(t) of the corresponding
singularity. The period map (cf. section 12.2)
DpMHS, t н> F'(t)
A3.29)
to a classifying space for polarized mixed Hodge structures on H°° is
holomorphic.
Denote the corresponding objects for / by H°°, Ш%, h, S, and F*G) for
7 e S/j.. Choose any isomorphism
X : (Я|°, A, S) -> (fl|°, h, S).
One obtains a holomorphic period map
' PMHS
The coordinate change <p-t induces
(pitCoh :
: #°
A3.30)
A3.31)
tf~ which
maps F*(f,-) to F*(^). Therefore x ° <Pi,coh e Gz = Aut(H^, h, S) acts on
Dpmhs with
X о <Р1,с
= $PMHs(ti).
A3.32)
Now first note that ФPMHs(ti)-+ *pmhs@) and ^pMHsffi)^-^pmhs@)
for г ->¦ oo, and second that the group Gz acts properly discontinuously on
Dpmhs (Theorem 10.22). Therefore there exists an infinite subset / с N and
an isomorphism cpcoh : H°° -* Я°° with
and
= <Pcoh for i e I
<Pcoh(F'@)) - F @).
A3.33)
A3.34)
So / and f have isomorphic polarized mixed Hodge structures. (One can
apply the same arguments to DBL instead of DPMhs- If the global Torelli type
conjecture 12.7 were proved one could stop here.)
Choose a monodromy invariant filtration U. on H°° which is opposite to
F*@) and a generator y\ of the 1-dimensional space С-г^_а|]Я°?2„,в1. By
Theorem 11.1 the pair ([/., yi) induces a flat metric g on M.

1]
2
:=!
a \
244
Moduli spaces for singularities
The filtration U. is opposite to F'(t) for all t 6 5M (sufficiently close to 0).
The construction in section 11.1 gives the same metric g for all germs (A/, t)
with t e Бц.
The image (?/., у i) under ^c0/, satisfies the same properties with respect to
F (t) for t e 5M and induces a flat metric g on A/.
By Lemma 13.6 (b), the maps cpi<M, i e I, are isomorphisms of germs of
Frobenius manifolds
<Pi,M-({M,ti),o,e,E,g)-+ ((M,7i),o,e,E,g). A3.35)
Even more, by combining Lemma 13.4 and Lemma 13.6 (b) one sees that they
differ at most by translations (with respect to the flat structures).
Thus for sufficiently large i € I, the map ц>^м extends to a neighbourhood
of 0 e M, and <pi,m@) —> 0 for i -*¦ oo.
If onehad^,?w@)/Oforanarbitrarilylarge/ 6 /thentheset{^,,«@) \i € I]
would cluster around 0 e M. The germs ((Л/, <р,-,м@)), о, ?, E) of F-manifolds
with Euler field would all be isomorphic. This is not possible by Corollary 4.16.
(Also, the singularities for the parameters <р,,л/@) б 5М would all be right
equivalent. With the proof below of Theorem 13.15 and with a closer look at
the action of the algebraic group jklZ on the algebraic variety C(k, f) one finds
that this is also not possible.)
Therefore <р,,м@) = 0 for large i e I. This is the limit isomorphism from
A3.28). One finishes the proof with Theorem 13.2 (b) or Scherk's result. D
Theorem 13.18 Let f : (C+1, 0) -* (C, 0) be an isolated hypersurface sin-
singularity. There exists a representative M of the base space (A/, 0) of a semiu-
niversal unfolding of f with the following properties.
The finite group hut м '¦= Aut((A/, 0), о, е, Е) of automorphisms of the germ
of an F-manifold acts on M. Ift and t are in the (i-constant stratum S^CZM
and (p : ((A/, t), о, е, Е) —> ((A/, t), о, е, Е) is an isomorphism of germs of
F-manifolds, then <p e Aut«.
Proof. We may suppose n > 1. Assume that such a representative does not
exist. Then one can choose (for some representative M) two sequences (f,),?N
and (f|)ieN with f,-, f,- 6 S^, f,- —>• 0 and t-, —> 0 for i —*¦ oo, and a sequence of
isomorphisms
((A/, ti), o, e, E) -* ((А/,?)), о, е, Е)
A3.36)
with cpi g Autw for any i.
By Theorem 13.2 (b), for each i the singularities which correspond to Ц
and t, are right equivalent and there exists a coordinate change between them
13.3 Global moduli spaces for singularities
245
which induces щ. But then the proof of Theorem 13.17 provides an infinite
subset / С N such that all щ for i e I are equal and contained in AutM. A
contradiction. D
Remark 13.19 This resultdoes not extend to t, t e M — 5M. For example, in the
case of an A^-singularity, the Lyashko-Looijenga map (section 3.5 and E.49))
M -+ ©* is finite of degree (/x + 1)д~'. This implies that a generic germ of a
semisimple F-manifold with Euler field turns up at (д + 1)M~' different points
in M. But the group Aut^ only has order jjl + 1. So most of the isomorphisms
of the germs of F-manifolds do not extend to 0 e M.
Proof of Theorem 13.15: The jt7^-orbit of jkf in m2/m*+1 has codimension
/i - 1. A transversal disc for / is an unfolding F : (C"+1 x M, 0) -* (C, 0)
of / with the following properties:
M С См~' is a neighbourhood of 0;
for any parameter a = (au ¦ ¦ ¦, g^-i) 6 M the function germ Fa is
Fa = F\(C"+l x {a}, @, a)) 6 m2; A3.37)
of course Fq = /;
the map Tk:M-+ m2/m*+l, а н> jkFa is an embedding and the image
Tk(M) intersects jklZ ¦ jkf transversally in jkf.
If we choose a smooth germ (R, id) С (JkR-,id) which is transversal to the
stabilizer of jk f then the natural map of germs
(R, id) x (Tk(M) П C(k, /), jkf) -* (C{k, /), jkf) A3.38)
is an isomorphism. We need the germ (Tk(M) П C(k, /), jkf) to be isomorphic
to the ^-constant stratum EД, 0) С (А/, 0) of /. It will follow from results of
Gabrielov and Teissier.
Gabrielov [Ga] constructed explicitly a semiuniversal unfolding F : (C"+l x
A/, 0) -» (C, 0) of /, a transversal disc F for /, and an embedding
(|, т) = (?o, ¦ • ¦, I,,, t,, ..., v> : (A/, 0) -»¦ (C+1 x M, 0) A3.39)
with
). A3.40)
The critical space (C, 0) С (C+1 xM, 0) (cf. section 10.1)and the intersection
С П F-'@) are smooth, С = prM(C П F~40)) С М is the discriminant.
Gabrielov's construction yields an isomorphism
(§, r): (M, 0) ^ (СПГ'@),0) A3.41)

246
Moduli spaces for singularities
3

&
б
which maps T^{Tk{M) П C(Jfc, /), j*/) to the preimage in (С П F-'(O), 0) of
the д-constant stratum EM, 0).
The projection to the /j,-constantstratum is clearly ahomeomorphism. Teissier
[Tel, §6] showed that it is an isomorphism. He used the fact that the map
CnF"'@) —> V is isomorphic to the development of the discriminant (cf. for
example section 3.5) to construct a section (M, 0) -» (C"+1 x M, 0) which
contains the preimage ofE^, 0) in (С П F~'@), 0).
Therefore the germs {T^{M) П C(&, /), j*/) and EM, 0) are isomorphic.
Now Theorem 13.17 shows that the quotient topology of M^ = C(k, f)/jkR.
is Hausdorff. This is the first of two conditions in a criterion of Holmann
[Hoi, Satz 17] for the existence of an analytic geometric quotient. The second
condition is the existence of holomorphic functions in a neighbourhood of j^f
in C{k, /), which are constant on the y'tTJ.-orbits and which separate points in
different orbits.
By A3.38) and the above isomorphism of germs it is sufficient to show the
existence of such functions on 5M. But in that case itfollows from Theorem 13.18
and from the classical result of Cartan [Ca] that a quotient by a finite group is
a reduced complex space. This shows parts (a) and (b) in Theorem 13.15.
The canonical complex structure on 5Д from Theorem 12.4 is invariant un-
under Aut«, for example because these automorphisms respect the Frobenius
manifold structure from t/^0' and the canonical complex structure is com-
completely determined by this Frobenius manifold structure A st step in the proof of
Theorem 12.4). One can apply locally [Каир, 49 A. 16] and glue the complex
structures. This gives part (c). ?
Remark 13.20 (a) Consider the local period map Ф : S^ —> Dbl for a singu-
singularity /. If <p € TV is a symmetry then the action of cpM on 5M С М and the
action of (pcof, on ФEМ) С Dbl are compatible. Now Theorem 13.15 shows
that the global period map Мц -» DBL/Gz is a map between reduced complex
varieties.
(b) A part of the splitting lemma says that two singularities f(xo,..., xn)
and g(xo,..., xn) are right equivalent if and only if / + x^+x and g + x%+l
are right equivalent (e.g. Remark 5.7 (i)). Therefore the global moduli space
М„(/) embeds into M^f + x*+l).
It would be very desirable to know the answer to the following very weak
form of the Zariski conjecture. Do the singularities in a д-homotopy class have
either all multiplicity 2 or all multiplicity > 3?
If the answer is yes, M^if) and Мц(/ + х%+1) are isomorphic, if no, the
second one would be larger in some cases.
13.3 Global moduli spaces for singularities
247
(c) The answer to the question is yes for semiquasihomogeneous singularities
and for curve singularities.
An interesting related result has been shown by Navarro-Aznar [Nav]: sup-
suppose f(x0, ...*„) and g(xo,..., х„) are singularities with the same topological
type and suppose that the rank of the Hessian of /, rank( g^r)(O), is odd (this
implies that its multiplicity is 2). Then also the multiplicity of g is two.
This implies that two surface singularities with the same topological type
both have multiplicity two or both have multiplicity > 3.
(But precisely in the surface case it is not yet clear whether д-homotopy
implies the same topological type, cf. [LeR]).
(d) If the answer is yes for some д-homotopy class then the groups Rf for
singularities / in this д-homotopy class satisfy a semicontinuity property. Then
by Theorem 13.18 and Theorem 13.9 the groups RFl for the singularities in the
/j,-constant stratum of a fixed singularity / are (isomorphic to) subgroups of Rj.
For semiquasihomogeneous singularities the semicontinuity property of Rf
has been proved previously in [Mue2].

Chapter 14
Variance of the spectral numbers
Section 14.3 gives a surprising statement on the spectral numbers of quasi-
homogeneous singularities. It comes from properties of the G-function of a
semisimple or massive Frobenius manifold. General remarks, the definition
of Dubrovin, Zhang, and Givental and some properties of it are presented in
section 14.2. The socle field of a Frobenius manifold is related to the simpler
part of this G-function. It is more or less known, but not treated systematically
in the literature. This is provided in section 14.1.
14.1 Socle field
A Frobenius manifold has another distinguished vector field besides the unit
field and the Euler field. It will be discussed in this section. We call it the socle
field. It is used implicitly in [Du4, Theorem 1.1] and [Gi7].
Let (M, o, e, g) be a manifold with a multiplication о on the tangent bundle,
with a unit field, and with a multiplication invariant metric g. We do not need
flatness and potentiality and an Euler field in the moment.
Each tangent space T,M is a Frobenius algebra and splits uniquely into a
direct sum of Gorenstein rings (cf. section 2.1)
/@
k=\
with maximal ideals m,,* С (TtM\ and units ek such that e
and thus
(T,M)jo(T,M)k =
g((T,M)j,(T,M)k)={0) for ]фк.
248
A4.1)
A4.2)
A4.3)
14.1 Socle field
249
The socle Ann(r,W)l(m,,t) is 1-dimensional and has a unique generator H,k
which is normalized such that
g(ek, H,,k) = dim(T,M)k.
A4.4)
The following lemma shows that the vectors ^t H, k glue to a holomorphic
vector field, the socle field of (M, o,e,g).
Lemma 14.1 For any dual bases X\,..., Xm and X\,..., Xm of T,M, that
means, g(X,-, Xj) = <5,;, one has
'(О
Е
*=i
A4.5)
Proof. One easily sees that the sum J2 X,- о X,- is independent of the choice of
the basis X\,... ,Xm. One can suppose that l(t) = 1 and that X\,...,Xm are
chosen such that they yield a splitting of the filtration T,M D mt,\ Э m^, Э
Then g(e, X, о X,) = 1 and
g(m,,i, X,- о X,) = g(X, о mM, X,-) = 0.
ThusX,oX, = ?#,,,. D
It will be useful to fix the multiplication and vary the metric.
Lemma 14.2 Let (M, о, e, g)bea manifold with multiplication о on the tangent
bundle, unit field e and multiplication invariant metric g.For each multiplication
invariant metric g there exists a unique vector field Z such that the multiplication
with it is invertible everywhere and for all vector fields X, Y
g(X, Y) = g(Z о X, Y).
The socle fields H and H of g and g satisfy
H = ZoH.
A4.6)
A4.7)
Proof. The situation for one Frobenius algebra is described in Lemma 2.2.
It yields A4.6) immediately. Formula A4.7) follows from the comparison of
A4.4) and A4.6). D
Denote by
X i-> H о X
A4.8)

250
к
t
I
z
• -oil ;
Or
И ! Ir
Variance of the spectral numbers
the multiplication with the socle field H of (M, o, e, g) as above. The socle
field is especially interesting if the multiplication is genetically semisimple,
that means, generically l(t) = m. Then the caustic K. = {t e M \ l(t) < m) is
the set where the multiplication is not semisimple. It is the hypersurface
/C = det(#opr'@). A4.9)
In an open subset of M — /C with basis e\,..., em of idempotent vector fields
the socle field is
l
-e;.
A4.10)
It determines the metric g everywhere because A4.10) determines the metric
at semisimple points.
If (M, o, e) is an F-manifold then each germ ((M, t), o, e) decomposes uni-
uniquely into a product of irreducible germs of F-manifolds (Theorem 2.11). This
extends the infinitesimal decomposition A4.1).
At semisimple points one has the product А"г if m 1-dimensional germs of
F-manifolds. The 2-dimensional irreducible germs are classified in Theorem
4.7, those with generically semisimple multiplicationformaseries/2(n), n > 3,
with /2C) = A2.
Theorem 14.3 Let (M, o, e, g) be a massive F-manifold with multiplication
invariant metric g. Suppose that at generic points of the caustic the germ of the
F-manifold is of the type I2(n)A"'~2.
Then the function det(Hop) vanishes with multiplicity n —2 along the caustic.
Proof. The manifold M = O" with coordinate fields 5,- = ^- and multiplication
defined by
5) о &2 =
t
8\,
A4.11)
A4.12)
A4.13)
5,- о 8j = вув, if (i, j) i {A, 2), B, 1), B, 2)}
is an F-manifold with a global decomposition С2 к С x • • • x С of the type
/гСяМ^2-^61111^6^ for the components are 5|, <$з> ¦ ¦ • > 5,„, the global unit
field is e = 8\ + S3 -\ + ?„,, the caustic is /C = {t \ t2 — 0}. The idempotent
vector fields in a simply connected subset of M — /C are
1.1
2'
e, = 5,- for i > 3,
e\/i = -Si ± -t2 2 82,
A4.14)
A4.15)
14.2 G-function of a massive Frobenius manifold 251
canonical coordinates there are
2 "
mi/2 = tx ± -q, A4.16)
Ui=tj for«>3. A4.17)
A multiplication invariant metric g is uniquely determined by the 1-form e =
g(e,.). Because of A4.7) it is sufficient to prove the claim for one metric. We
choose the metric with 1-form
eE,) = l-5n. A4.18)
The bases 5i, 82, 53,..., 5m and 82, 8\, 53,..., 8m are dual with respect to this
metric. Its socle field is by Lemma 14.1
H = 282 + 83 + ¦ ¦ ¦ + 8П:
A4.19)
2
and satisfies det(#op) =
If an F-manifold M has at generic points of the caustic germs of the type
I2(n)A™~2 then the set of nongeneric points is empty or has codimension > 2
inM. ?
14.2 G-function of a massive Frobenius manifold
Associated to any simply connected semisimple Frobenius manifold is a fas-
fascinating and quite mysterious function. Dubrovin and Zhang [DuZl][DuZ2]
called it the G-function and proved the most detailed results for it. But Givental
[Gi7] studied it slightly earlier, and it originates in much older work. It takes
the form
G(t) =\ogr,-± log J A4.20)
and is determined only up to addition of a constant. First we explain the sim-
simpler part, log J. Let (M, о, е, Е, g) be a semisimple Frobenius manifold with
canonical coordinates u\,...,um and flat coordinates t\,...,tm. Then
/ = i
¦ constant
A4.21)
is the base change matrix between flat and idempotent vector fields.
One can rewrite it with the socle field. Denote щ := g(e,-, e,) and consider
the basis v\,..., vm of vector fields with
1 I
= —— e,-
A4.22)

fii
3 ... e
252
Variance of the spectral numbers
(for some choice of the square roots). The matrix det(g(i>,-, Vj)) = 1 is constant
as is the corresponding matrix for the flat vector fields. Therefore
constant ¦ / =
i — det(#op)
5
A4.23)
Here H — 53 Vj о и,- is the socle field.
One of the origins of the first part log r/ is the geometry of isomonodromic
deformations. The second structure connections and the first structure connec-
connections of the semisimple Frobenius manifold are isomonodromic deformations
overP1 x M of restrictions to a slice P1 x {r}. The function Г/ is their г -function
in the sense of [JMMS][JMU][JM][Mal4]. See [Sab4] for other general ref-
references on this.
The situation for Frobenius manifolds is discussed and put into a Hamiltonian
framework in [Du3, Lecture 3], [Man2, II§2], and in [Hi]. The coefficients #,¦
of the 1-form dlog T/ = ? Hjdiij are certain Hamiltonians and motivate the
definition of this 1-form. Hitchin [Hi] compares the realizations of this for the
first and the second structure connections.
Another origin of the whole G-function comes from quantum cohomology.
Getzler [Ge] studied the relations between cycles in the moduli space M. \ ,4 and
derived from it recursion relations for genus one Gromov-Witten invariants of
projective manifolds and differential equations for the genus one Gromov-
Witten potential.
Dubrovin and Zhang [DuZl, chapter 6] investigated these differential equa-
equations for any semisimple Frobenius manifold and found that they have always
one unique solution (up to addition of a constant), the G-function, a function
which was proposed in [Gi7].
They also proved the major part of the conjectures in [Gi7] concerning G(t).
Finally, they found that the potential of the Frobenius manifold (for genus zero)
and the G-function (for genus one) are the basements of full free energies in
genus zero and one and give rise to Virasoro constraints [DuZ2]. Givental found
formulas for the full free energies at higher genus [,Gi8] and proved Virasoro
constraints for them [Gi9]. Exploiting this for singularities will be a big task
for the future.
For our application in section 13.3 we need only the definition of log T/ and
the behaviour of G(t) with respect to the Euler field and the caustic in a massive
Frobenius manifold. We have to summarize some known formulas related to
the canonical coordinates of a semisimple Frobenius manifold ([Du3], [Man2],
also [Gi7]).
14.2 G-function of a massive Frobenius manifold 253
The 1-form e = g{e,.) is closed and can be written as e = dr\. One defines
j)i := ejT) - g{et,e) = g(eit e,),
raj :=e,-e/4 = ejTjj =е}гц,
Vjj :=-(«; -Uj)yij,
l ¦
dlogr/ := -
n
A4.24)
A4.25)
A4.26)
A4.27)
A4.28)
A4.29)
Theorem 14.4 Let (M, o,e, E, g) be a semisimple Frobenius manifold with
global canonical coordinates u\, ...,um.
(a) The rotation coefficients Yij (for i Ф j) satisfy the Darboux-Egorojf
equations
ekyij = MYkj &гкф1ф]фк, A4.30)
еуц = 0 fori^j. A4.31)
(b) The connection matrix of the flat connection for the basis v\, ...,vm
from A4.22) is the matrix {yi}d{uj - uj)). The Darboux-Egorojf equations are
equivalent to the flatness condition
d(yud(Ui - uj)) + (y,7d(«; - uj)) л (Yijd(uj - Uj)) = 0. A4.32)
(c) The 1-form d log Г/ is closed and comes from a function log r,.
(d) The Euler field E satisfies Е(гц) = (D - 2)гц and E{yu) = -yu.
(e) If the canonical coordinates are chosen such that E = ? u'e' tlten tlte
matrix -(V,j) is the matrix of the endomorphism § i T ih
respect to the basis V\, ..., vm.
VE-§ id on TM with
Proof, (a) and (b) See [Du3, pp. 200-201] or [Man2, I§3].
(c) This can be checked easily with the Darboux-Egoroff equations.
(d) It follows from Lie?(g) = D ¦ g and from [e,-, E] = e,.
(e) This is implicit in [Du3, pp. 200-201]. One can check it with (a) and (b)
and(d). D
The endomorphism V is skewsymmetric with respect to g and flat with eigen-
eigenvalues d\ - f; the numbers dt can be ordered such that d\ = 1, dj + dm+1 _,• = D
(cf. Remark 9.2 (e)).

Т'П
пт-
пт"il
1..
32.'
3"
"п..
¦ '' :,|i !;
1 Vj
i ' ¦
254 Variance of the spectral numbers
Corollary 14.5 [DuZl, Theorem 3] Suppose that E — J2ще-,. Then
? log т/ =-i
48
Proof:
A4.33)
=: у. A4.34)
A4.35)
Formula A4.23) shows E(J) = ,
tion of the G-function.
i^=2 J. Now A4.34) follows from the defini-
D
If M is a massive Frobenius manifold with caustic /C, one may ask which
kind of poles the 1 -form d log t/ has along /C and when the G-function extends
over 1С.
In [DuZl, chapter 6] the G-function is calculated for the 2-dimensional
Frobenius manifolds h{n), n > 3, on M — C2 with coordinates (t\, t2) and
e = —-. It turns out to be
1 B-иХЗ-и)
A4.36)
Especially, for the case /гC) = A2 the G-function is G(t) = 0. This was also
checked in [Gi7]. Givental concluded that in the case of singularities the
G-functionof thebase space of a semiuniversal unfolding with some Frobenius
manifold structure extends holomorphically over the caustic. This is a good
guess, but it does not follow from the case Аг, because a Frobenius manifold
structure on a germ of an F-manifold of type A2^T~2 for ш > 3 is never
the product of the Frobenius manifolds Аг and A™~2 (the numbers d\,..., dm
would not be symmetric). However, it is true, as the following result shows.
Theorem 14.6 Let (M, о, е, Е, g) be a simply connected massive Frobenius
manifold. Suppose that at generic points of the caustic K. the germ of the
underlying F-manifold is of type l2(n)A™~2 for one fixed number n > 3.
14.2 G-function of a massive Frobenius manifold
255
(a) The form dlog Г/ has a logarithmic pole along /C with residue — (62[-
along K.reg.
(b) The G-function extends holomorphically over K, if and only ifn = 3.
Proof. Theorem 14.3 and A4.23) say that the form — ^d log J has a logarithmic
pole along/C with residue ^ along ICreg. This equals ^f^f- if and only ifn = 3.
So (b) follows from (a).
It is sufficient to show (a) for the F-manifold M = Cm in the proof of Theorem
14.3, equipped with some metric which makes a Frobenius manifold out of it
(we do not need an Euler field here). Unfortunately we do not have an identity for
dlog Г/ as A4.7) for the socle field which would allow only a most convenient
metric to be considered.
We use A4.11)-A4.17) and A4.24)-A4.29) and consider a neighbourhood
of 0 6 Cm = M. Denote for j > 3
П% Ъ
T2J := («i - «y)-^- - ( - иу)
n2
A4.37)
With
) ф 0 for j > 3, A4.29) and A4.16) one calculates
8dlogt/ — holomorphic 1-form + T\2
tnf
+ ? T2jtfdt2 - Y, Tijduj. A4.38)
From A4.14) one obtains
1
у>з
2
у>з
1
-n+21
1 n-2
4 2
A4.39)
A4.40)
A4.41)
The vector 52|o is a generator of the socle of the subalgebra in TqM which
corresponds to I2(n). Therefore 52(^)@) # 0. It is not hard to see with A4.39)

1
'Г! Я
256
Variance of the spectral numbers
257
and A4.16) that the terms T{j and T2j t2 2 for j > 3 are holomorphic at 0. The
term 7"i2 is
T\2 = - •
= 8 .
n
• 1- holomorphic 1-form. A4.42)
In t2
This proves part (a).
?
Remarks 14.7 (a) The base spaces of semiuniversal unfoldings meet the case
n = 3 in Theorem 14.6. Closely related are base spaces of certain unfoldings
of tame functions. The germs of F-manifolds are isomorphic to products of
the germs of F-manifolds from hypersurface singularities. But Sabbah [Sab3]
[Sab2][Sab4] equipped them with a metric such that the Frobenius manifold
structure is in general not a product (cf. section 11.4).
(b) It might be interesting to look for massive Frobenius manifolds which
meet the case n = 3 in Theorem 14.6, but where the underlying F-manifolds
are not locally products of those from hypersurface singularities. In view of
Theorem 5.6 the analytic spectrum of such F-manifolds would have singulari-
singularities, but only in codimension > 2, as the analytic spectrum of Аг is smooth.
The analytic spectrum is Cohen-Macaulay and even Gorenstein and a
Lagrange variety. P. Seidel (Ecole Polytechnique) showed me a normal and
Cohen-Macaulay Lagrange surface. But it seems to be unclear whether there
exist normal and Gorenstein Lagrange varieties which are not smooth.
14.3 Variance of the spectrum
By Theorem 14.6 the germ (M, 0) of a Frobenius manifold as in Theorem 11.1
for an isolated hypersurface singularity / has a holomorphic G-function G(t),
unique up to addition of a constant. By Corollary 14.5 and Theorem 11.1 this
function satisfies
III ГУ .. ГУ1 1
y. A4.43)
-a,)
48
So it has a very peculiar strength: it gives a hold at the squares of the spectral
numbers ai,..., aM of the singularity. Because of the symmetry a,- +aM+i_/ =
n — 1, the spectral numbers are scattered around their expectation value '¦—-.
One may ask about their variance - Yli=\(.ai ~ ^y^J-
14.3 Variance of the spectrum
Conjecture 14.8 The variance of the spectral numbers of an isolated hyper-
hypersurface singularity is
or, equivalently,
-I
Y >
Theorem 14.9 In the case of a quasihomogeneous singularity f
n-Г2
and
y=0.
A4.44)
A4.45)
A4.46)
A4.47)
Proof. One has the isomorphism (G/J/, mult., [/]) = (TqM, o, E\q). Here
/ € Jf and ?|o = 0 and therefore E G(t) =0. ?
Remarks 14.10 (a) When I presented Theorem 14.9 at the summer school on
singularity theory in Cambridge in August 2000, I asked for an elementary
proof of it. This was found by A. Dimca. It uses the characteristic function
of the spectral numbers. The variance is
A4.48)
A4.49)
In the case of a quasihomogeneous singularity with weights wo, ...,wn 6 @, |]
and degree 1 the characteristic function is
A4.50)
as is well known. Using this product formula A. Dimca [Di] showed that in the
case of a quasihomogeneous singularity the variance is ?"_0 ^-yj^ = a"l~2"' ¦ He
also made a conjecture dual to Conjecture 14.8 forthe case of tame polynomials:
there the inverse inequality to A4.44) should hold. The conjectures intersect in
the case of quasihomogeneous singularities and give there the equality A4.46).

f
m
'IP!' ¦
258
Vfan'ance of the spectral numbers
(b) M. Saito in September 2000 proved Conjecture 14.8 in the case of irre-
irreducible plane curve singularities [SM5].
(c) Т. Brelivet in May 2001 proved it in the case of plane curve singularities
with nondegenerate Newton polyeder [Bre].
(d) The only unimodal or bimodal families of not semiquasihomogeneous
singularities are the cusp singularities Tpqr and the 8 bimodal series. The spectral
numbers are given in [AGV2]. One finds
with equality only for the simple elliptic singularities. In the case of the 8
bimodal families one obtains
У = -rf~ • (l j—) > 0 A4.52)
48-к \ p + kJ
with к :— 9, 7, 6, 6, 5 for ?з,р, Z\ p, Q2,P, W\p, S\iP, respectively, and
Y =
48-at
A4-53>
with к := 6, 5, § for w\ p, S\p, U\tP, respectively.
(e) At the summer school in Cambridge in August 2000 Conjecture 14.8
was confirmed for many other singularities using the computer algebra system
Singular and especially the program of M. Schulze for computing spectral
numbers, which is presented in [SchuSt].
(f) In [SK8] K. Saito studied the distribution of the spectral numbers and
their characteristic function Xf heuristically and formulated several questions
about them. The G-function might help these problems to be continued.
(g) One can speculate that Conjecture 14.8, if it is true, comes from a more
profound hidden interrelation between the GauB-Manin connection and po-
polarized mixed Hodge structures. The existence of Frobenius manifolds and
G-functions alone is not sufficient, as the following shows.
In Remark 11.7 (b) an example of M. Saito [SM3,4.4] is sketched which leads
for the semiquasihomogeneous singularity / = x6 + y6 + x4y4 to Frobenius
manifold structures with {d\,.., d^} ф {1+ai—a,- \i = 1,..., /x}. The number
у in that case is у = — ¦— < 0.
(h) In the case of the simple singularities Ak, Дь E$, E-j, E$, the parame-
parameters 11,..., ?M of a suitably chosen unfolding are weighted homogeneous with
positive degrees with respect to the Euler field. Therefore G = 0 in these cases
(cf. [Gi7])
14.3 Variance of the spectrum
Lemma 14.11 The number у of the sum f(xo,..., x,,) + g(yo, ¦
singularities f and g satisfies
y(f + g) =
y(g)
¦ y(f).
259
¦, Ут) of two
A4.54)
Proof: Let ai,..., or^/) and /3i,..., /3M<g) denote the spectral numbers of /
and g. Then the spectrum of / + g as an unordered tuple is [AGV2][SchSt]
(a,- + pj + 1 | i = 1,..., fj.(f), 7 = 1,..., ii(g)). A4.55)
This and the symmetry of the spectra yields A4.54). D
Remarks 14.12 For any Frobenius manifold the variance ~ Ya=Mi ~ fJ
of the eigenvalues d\,...,dm of V? is interesting. It turns up not only as in
Corollary 14.5 related to the G-function in the semisimple case, but also in the
operator Lq of the Virasoro constraints in [DuZ2, B.30)] for any Frobenius
manifold.
Prior to [DuZ2] the Virasoro constraints were postulated in the case of quan-
quantum cohomology of projective manifolds with hp-q = 0 for p ф q in [EHX].
There a formula for the variance was considered which turned out to be a special
case of the following formula from [LiW] (cf. also [Bori]), which is valid for
any projective manifold:
p.q
A4.56)
Here c\ is the /th Chern class of the manifold, n is its dimension. The proof
uses the Hirzebruch—Riemann—Roch theorem. The formula is generalized to
projective varieties with at most Gorenstein canonical singularities in [Bat].
Comparing the right hand side with the singularity case, one can speculate
n ~ aM — ori, с„ ~ ц. and ask about ^с\сп_\ ~?.

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1
к
к

Index
Н°°, ПО, 136, 189, 196
Н", 168
«№) ciVtt, 168
Пт, 172, 174
Щ, 171, 188, 191
д-constant family, 193
д-constant stratum, 53, 218
д-homotopy class, 241
т-function, 252
analytic spectrum, 11, 24,56
automorphism group, 52
bifurcation diagram, 36,161
Birkhoff problem, 120, 214,216
boundary singularity, 69
Brieskorn lattice, 114, 172, 188
canonical coordinates, 18, 160
canonical extension, 142
caustic, 13,36, 161
classifying space for Brieskorn
lattices, 194
classifying space for PMHSs, 188
cohomology bundle, 168
covariant derivative, 20, 146
Coxeter group, 75,83
critical space, 62
Darboux-Egoroff equations, 253
de Rham cohomology, 174
deformed flat coordinates, 205
Deligne's /''¦«, 184
development, 30, 41, 246
discriminant, 36,40,47, 161
discriminant of a singularity, 66, 167
eigenspace, 111
eigenspace decomposition, 10
elementary part, 112
elementary section, 110, 136
Eulerfield, 14,25,29, 146
exhaustive filtration, 117
F-manifold, 14
filtration, 114, 115, 117
first structure connection, 154, 205
flat coordinates, 147
flat metric, 83
flat vector bundle, 109
Fourier-Laplace transformation, 156,
214,216
free divisor, 47, 134
Frobenius algebra, 10
Frobenius manifold, 22, 83, 146
front, 30
G-function, 251
GauB-Manin connection, 170
GauB-Manin system, 179, 181
Gelfand-Leray form, 171
generalized Milnor fibration, 168
generating family, 59, 68
generating function, 30, 36
good section, 122
Gorenstein ring, 10
Gromov-Witten invariants, 252
Grothendieck residue, 180
group of symmetries, 235
higher residue pairings, 180, 191
hypersurface singularity, 62, 165
infinitesimal Torelli type result, 226
intersection form, 150, 153, 181, 189
isolated hypersurface singularity, 62,
165
isomonodromic deformations, 252
269

, 62
i|iMiiiiiiii(iilu<in<>lii|iy, -II. 252
KcHlniru Spencer niiip, 62. 166
Uigmnyc tihraiion. 31
I.agrange map, 31
Lagrange variety, 24
lattice, I 13, 115
Lefschetz thimble, 215
Levi-Civita connection, 146
Lie derivative, 14, 146
logarithmic differential form, 131
logarithmic pole, 118, 134, 158, 162
logarithmic vector field, 47, 131
Lyashko-Looijenga map, 30, 36, 55, 80
M-tame function, 213, 217
massive, 24
massive Frobenius manifold, 160
metric, 145
microdifferential operator, 113, 174
Milnor fibration, 168
miniversal Lagrange map, 33
mirror symmetry, 211
mixed Hodge structure, 184, 193
modality, 53
moderate growth, 112
moduli of germs of F-manifolds, 93
moduli space Мц, 225, 241
monodromy, 110, 162, 189
monodromy group, 153, 168, 204
multiplication invariant, 10, 21, 146
multivalued section, 110
normal crossing case, 132, 140
open swallowtail, 95
opposite filtration, 122, 185, 197
order, 112
oscillating integral, 157, 214, 216
period map, 225
Picard-Lefschetz transformation, 153, 168
PMHS, 184
Poincare rank, 134
polarized mixed Hodge structure, 184, 192
polarizing form, 184
pole of order < r + 1, 134
potential, 22, 147
potentiality, 22, 146
primitive form, 104, 178, 202
primitive subspace, 183, 185
principal part, 112
reduced Kixlinni Spencer шар, 62, 163
reduced l.ynshko-l.<xiijcn};ii map, 30,
M>
rellexive, 133
reflexive extension, 133, 139
regular singular. 143
residual connection. 135. 137. 158
residue endoniorphism, 114, 135, 137,
158, 162
restricted bifurcation diagram, 38, 50,
51
restricted caustic, 38
restricted Lagrange map, 34
Riemann-Hilbert problem, 120
Riemann-Hilbert-Birkhoff problem.
121
right equivalent, 64
saturated lattice, 116, 118
second structure connection, 149, 204
semisimple, 10
semiuniversal unfolding, 63, 165
simple F-manifold, 55, 77
small quantum cohomology, 211
smooth divisor, 132
socle field, 249
spectral number, 114, 128, 193, 256
spectral pair, 193
spectrum of a Frobenius manifold,
84, 147
spectrum of a singularity, 172
splitting lemma, 67
stably right equivalent, 67
standard form, 43
strict morphism. 185
symmetries of singularities, 235
Torelli type conjecture, 225, 239
Torelli type result, 226
unfolding, 62
unit field, 14
V-filtration, 112
variance, 256
variation operator, 189
versal Lagrange map, 33, 90
versal unfolding, 63
Virasoro constraints, 252
weight filtration, 183