Valentine S. Kulikov
Moscow State University of Printing
Mixed Hodge Structures and Singularities
Cambridge
UNIVERSITY PRESS

Contents
Introduction page xi
I The Gauss-Manin connection 1
1 Milnor fibration, Picard-Lefschetz monodromy transforma-
transformation, topological Gauss-Manin connection 1
1.1 Milnor fibration 1
1.2 Cohomological Milnor fibration 1
1.3 Topological Gauss-Manin connection 2
1.4 Picard-Lefschetz monodromy transformation 2
2 Connections, locally constant sheaves and systems of linear
differential equations 3
2.1 Connection as a covariant differentiation 3
2.2 Equivalent definition: a covariant derivative along a
vector field 4
2.3 Local calculation of connections. Relation to differen-
differential equations 5
2.4 The integrable connections. The De Rham complex 6
2.5 Local systems and integrable connections 7
2.6 Dual local systems and connections 8
3 De Rham cohomology 10
3.1 The Poincare lemma 10
3.2 Relative De Rham cohomology 11
3.3 De Rham cohomology for smooth Stein morphisms 11
3.4 Coherence theorem 12
3.5 On the absence of torsion in the De Rham cohomology
sheaves 12
3.6 Relation between i?f(/*Q/) and f*J&P(Qy) 13
^ 4 Gauss-Manin connection on relative De Rham cohomology 14

VI
Contents
Contents
vn
4.1 Identification of sheaves of sections of cohomological
fibration and of relative De Rham cohomology 15
4.2 Calculation of the connection on a relative De Rham
cohomology sheaf 16
4.3 The division lemma. The connections on the sheaves
.%?PiR(X/S)forp^n-l 17
4.4 The sheaf 'JT = /*aj/5/d(/*Q^) 19
4.5 Meromorphic connections 20
4.6 The Gauss-Manin connection as a connecting
homomorphism 21
Brieskorn lattices 23
5.1 Brieskorn lattice ".Зё 24
5.2 Calculation of the Gauss-Manin connection V on 'Ж 25
5.3 Increasing filtration on .M@) 25
5.4 A practical method of calculation of the Gauss-Manin
connection 27
5.5 Calculation of the Gauss-Manin connection of quasi-
homogeneous isolated singularities 28
Absence of torsion in sheaves Зв(~Ъ of isolated
singularities 30
6.1 The presence of a connection implies the absence of
torsion 30
6.2 A theorem of Malgrange 31
6.3 Connection on a pair (E, F) 32
6.4 Sheaves J&(~p> are locally free 1'* 32
Singular points of systems of linear differential equations 33
7.1 Differential equations of Fuchsian type 33
7.2 Systems of linear differential equations and
connections 34
7.3 Decomposition of a fundamental matrix Y{f) 35
7.4 Regular singular points 36
7.5 Simple singular points 36
7.6 Simple singular points are regular 37
7.7 Connections with regular singularities 39
7.8 Residue and limit monodromy 41
Regularity of the Gauss-Manin connection 42
8.1 The period matrix and the Picard-Fuchs equation 42
8.2 The regularity theorem follows from Malgrange's
theorem 44
8.3 The regularity theorem and connections with
logarithmic poles 44
The monodromy theorem 46
9.1 Two parts of the monodromy theorem 46
9.2 Eigenvalues of monodromy 47
9.3 The size of Jordan blocks 49
9.4 Consequences of the monodromy theorem.
Decomposition of integrals into series 49
10 Gauss-Manin connection of a non-isolated hypersurface
singularity 51
10.1 De Rham cohomology sheaves 51
10.2 Coherence 52
10.3 Relation between 3ffP(f*Qy) and /*J?f(Q/) 53
10.4 A general method of extension of a singular
connection over the whole disk 53
10.5 The sheaves 3@?„ and the Gauss-Manin connection
д,:Зё[_2)^.Жх) 54
10.6 The sheaves -M^-, and the Gauss-Manin connection
dt:M[_x)^M(^ , 56
10.7 A generalization of diagram E.3.4) " 57
П Limit mixed Hodge structure on the vanishing
cohomology of an isolated hypersurface singularity 60
1 Mixed Hodge structures. Definitions. Deligue's theorem 60
1.1 Pure Hodge structure 60
1.2 Polarised HSs 61
1.3 Mixed Hodge structure 61
1.4 Deligne's theorem 62
2 The limit MHS according to Schmid 62
2.1 Variation of HS: geometric case 62
2.2 Variation of HS: definition 63
2.3 Classifying spaces and period mappings 63
2.4 The canonical Milnor fibre 64
2.5 The Schmid limit Hodge filtration F's 67
2.6 An interpretation of F's in terms of the canonical
extension of Зё 69
2.7 The weight filtration of a nilpotent operator 70
2.8 Schmid's theorem 73
3 The limit MHS according to Steenbrink 73
3.1 The limit MHS for projective families: the case of
unipotent monodromy 74
3.2 The limit MHS for projective families: the general
case 75
3.3 Brieskorn construction 77
3.4 L.mit MHS on a vanishing cohomology 78
3.5 The weight filtration on Hn(Xoo). Symmetry of Hodge
numbers 79

4.2
4.3
4.4
4.5
5.2
5.3
5.4
viii Contents
4 Hodge theory of a smooth hypersurface according to
Griffiths-Deligne 82
4.1 The Gysin exact sequence . 82
Hodge theory for a complement U = X\Y. Hodge
filtration and pole order filtration 83
De Rham complex of the sheaf B[y]x and the
cohomology of a hypersurface Y 85
The case of a smooth hypersurface Yin a projective
space X=Pn+l ¦ 86
Generalization to the case of a hypersurface with
singularities 87
The Gauss-Manin system of an isolated singularity 88
5.1 Hodge theory of a smooth hypersurface in the relative
case 89
The Gauss-Manin differential system 90
Interpretation of the complex DRz/s (#[r]z) in terms
of the morphism /: X —> S 91
Connection between the differential system .3$ x and
the Brieskorn lattice Ж{Щ 94
Decomposition of a meromorphic connecf ion into a direct
sum of the root subspaces of the operator tdt. The F-
filtration and the canonical lattice 95
6.1 'Block'decomposition 95
6.2 Decomposition of a meromorphic connection .Ж into
a direct sum of the root subspaces 96
6.3 The order function a and the V--filtration - 98
6.4 Identification of the zero fibre of the canonical
extension SS and the canonical fibre of the fibration H 99
6.5 The decomposition of sections со е Ж into a sum of
elementary sections 100
6.6 Transfer of automorphisms from the Milnor lattice H
to the meromorphic connection УМ 101
The limit Hodge filtration according to Varchenko and to
Scherk- Steenbrink 103
7.1 Motivation of Scherk- Steenbrink's construction of the
Hodge filtration ' , 103
7.2 The definition of the limit Hodge filtration FSs
according to Scherk-Steenbrink 106
7.3 The Scherk-Steenbrink theorem 108
7.4 Varchenko's theorem about the operator of
multiplication by / in Q/ 110
7.5 The definition of the limit Hodge filtration F~ on
Hn(XO0) according to Varchenko 111
Contents ix
7.6 Comparison of the filiations Fss and FVa 111
7.7 Supplement on the connection between the Gauss-
Manin differential system JKx and its meromorphic
connection.^ 112
Spectrum of a hypersurface singularity 115
8.1 The definition of the spectrum of an isolated singularity 115
8.2 The spectral pairs Sppif) 117
8.3 Properties of the spectrum 118
8.4 The spectra of a quasihomogeneous and a semi-
quasihomogeneous singularity 119
8.5 Calculation of the spectrum of an isolated singularity
in terms of a Newton diagram 122
8.6 Calculation of the geometric genus of a hypersurface
singularity in terms of the spectrum 127
8.7 Spectrum of the join of isolated singularities 127
8.8 Spectra of simple, uni- and bimodal singularities 129
8.9 Semicontinuity of the spectrum. Stability of spectrum '
for ц-const deformations 130
8.10 Spectrum of a non-isolated singularity 132
8.11 Relation between the spectrum of a singularity with a
one-dimensional critical set and spectra of isolated
singularities of its Iomdin series 134
Ш The period map of a //-const deformation of an isolated
hypersurface singularity associated with Brieskorn
lattices and MHSs 139
1 Gluing of Milnor fibrations and meromorphic connections
of a /f-const deformation of a singularity 139
1.1 Milnor fibrations 140
1.2 Cohomological fibration 141
1.3 Canonical extension of the sheaf 3$ and the
meromorphic connection 142
2 Differentiation of geometric sections and their root
components wrt a parameter 144
2.1 Geometric sections and their root components 144
2.2 Formulae for derivatives of geometric sections and
their root components wrt a parameter 146
2.3 Decomposition of the root components of geometric
sections into Taylor series for upper diagonal deforma- M
tions of quasihomogeneous singularities 148
2.4 The sheaves Gri^@) 150

Contents
The period map
3.1 Identification of meromorphic connections in a
/г-const family of singularities
The period map defined by the embedding of
Brieskorn lattices
Example: the period map for En singularities
The period map for hyperbolic singularities Трл<г
The period map for simply-elliptic singularities
The period map defined by MHS on the vanishing
cohomology
The infinitesimal Torelli theorem
4.1 The V- -filtration on Jacobian algebra. The necessary
condition for /г-const deformation
Calculation of the tangent map of the period map. The
horizontality of the MHS-period map
The infinitesimal Torelli theorem
The period map in the case of quasihomogeneous
singularities
The Picard-Fuchs singularity and Hertling's invariants
5.1 The Picard-Fuchs singularity PFS(f) according to
Varchenko
The Hertling invariant Her\(f)
The Hertling invariants Her2(f) and Her3(f)
Hertling's results
3.2
3.3
3.4
3.5
3.6
4.2
4.3
4.4
5.2
5.3
5.4
References
Index
151
151
152
154
156
159
163
165
165
167
169
171
172
172
174
177
179
181
185
Introduction
The aim of this book is to introduce and at the same time to survey some
of the topics of singularity theory which study singularities by means of
differential forms. Here differential forms associated with a singularity are
the main subject as well as the main tool of investigation. Differential
forms provide the main discrete invariants of a singularity as well as
continuous invariants, i.e. they make it possible to study moduli of
singularities of a given type.
A singularity is a local object. It is a germ of an algebraic variety, or an
analytic space, or a holomorphic function. However, the majority of the
ideas and methods, used in the theory under consideration, originated in
the 'global' algebraic geometry. Therefore we first give a very brief and
schematic description of these ideas.
The idea of using differential forms and their integrals to define numeri-
numerical invariants of algebraic varieties goes back to the classic writers of
algebraic geometry. It will be important for us that holomorphic and
algebraic forms can be used to calculate the singular cohomology of a
smooth algebraic variety over C. Developing the ideas of Atiyah and Hodge
A955), Grothendieck A966) showed that H\Xan, С) ~ Н'Ш(Х/С), where
Н'Ш(Х/С) is the De Rham cohomology. Grothendieck defined Н'Ш(Х/С)
as the hypercohomology H^X, Q'x) of the complex of sheaves of holo-
holomorphic differential forms on X. The comparison theorem enables us to
calculate the cohomology of the complement to a hypersurface in projective
space by means of the cohomology classes generated by rational differential
forms.
Algebraic differential forms have also proved to be useful in the study of
the monodromy of a family of complex varieties, using the Gauss-Manin
connection. The monodromy transformation is the transformation of fibres

Xll
Introduction
(or their homotopic invariants) of a locally trivial fibration corresponding
to a loop in the base. This notion appears when studying the multivalued
analytic function, where it corresponds to the notion of the covering or
sliding transformation. Very often the monodromy transformation appears
in the following situation. Let /: X —> S be a proper holomorphic map of
an analytic space to the disk in the complex plane. Let Xt be the fibre
f~\t), t€S,S' = S\{0} and X' = f~\S'). By reducing the radius of S,
if necessary, we can make the fibration /': X' —> S' a locally trivial C°°-
fibration. The monodromy transformation Г associated with the loop in S'
surrounding 0 is called the monodromy transformation of the family / The
action of the monodromy Ton the vector space H*(X,) is obtained by the
parallel displacement of the cohomology classes in the fibres of the locally
constant fibration H= \_},е3'Н'{Х,, С) = R'f^CX'. Grothendieck also
defined the relative De Rham cohomology sheaves ^^(X/S) ~
R'f*(Qx/s). If /: X -> S is a smooth proper morphism of algebraic
varieties/C, then from Grothendieck's theorem it follows the existence
of the canonical isomorphism of coherent analytic sheaves
Жш(Х/Буп ~ «7*an(C) <g>c ^s». The presence of the locally constant
sheaf Я = R'f*(C) in J?'m(X/S) defines a topological connection on the
sheaf 3#'m(X/S). Katz and Oda A968) gave an algebraic definition of the
canonical connection (the Gauss-Manin connection) on the sheaves
J%'DR(X/S) such that the sheaves of its horizontal sections are R'f*(C).
They calculated this connection explicitly and showed that it reduces to the
definition orginally given by Manin A958) for the case in which X/S is an
algebraic curve over the field of functions.
The family of varieties X, defined by the morphism /degenerates at the
point OeS, and the Gauss-Manin connection has a singularity at the point
0. This singularity is regular. The notion of a regular connection generalizes
the classical notion of a differential equation with a regular singular point,
and is the subject of Deligne's book [Dl]. Katz gave an algebraic proof of
the regularity of the Gauss-Manin connection A970). Analytic proofs were
given by Griffiths A971) and by Deligne [Dl]. The regularity theorem is
related to the monodromy theorem. When the space X as well as all fibres
X,, t ф 0, is smooth, the monodromy theorem states that T is quasi-
unipotent on H*(Xt, Q), i.e. there exist positive integers к and N such that
(Tk — l)N = 0. There are several proofs of this basic theorem (Clemens
1969; Katz 1971; Griffiths & Schmid 1975). Many of the characteristic
features of the degenerate family/: X —> S become apparent in the proper-
properties of the monodromy. The monodromy of the family / is closely
connected with the mixed Hodge structure (MHS) on the cohomology
Introduction
Xlll
H*(X0) and H*(X,). The Hodge structure and the period map of a family
of algebraic varieties defined by the Hodge structure on the cohomology of
fibres H*(X,) are the second most important notions used in this book.
The concept of a pure Hodge structure is a formalization of the structure
of the cohomology groups of a compact Kahler manifold. From the theory
developed by Hodge in the 1930s it follows that H"(X, C) =
= Hq'p, where the cohomology vector space is identi-
identi®p+q=nHp'q,
fied with the vector space of harmonic forms, and harmonic forms, and
consequently the cohomology, can be decomposed into the direct sum of
components of the type (p, q). Under a variation of a variety X in a family
X, the variation of subspaces Hp'q(Xt) is not complex analytic. For this
reason it is more convenient to study the Hodge filtration
FpH"(X, C) = @r^pHr'n~r, which does depend analytically on the
parameter t. One can give an equivalent definition of the Hodge structure
in terms of the Hodge filtration. Using the isomorphism
H"(X, С) ~ HqR(X), we can express the Hodge filtration in terms of the
stupid (obvious) filtration o^pQ'x = {0 -» Q.px —> Qx+i -+...} on the
De Rham complex. One of Griffiths's discoveries [Gr] was that the Hodge
filtration is related to the pole order filtration: if со has a pole of order not
greater than к + 1 along a non-singular projective variety X, then the
residue Res со is a sum of forms of the type (p, q) with q =? k. This gives a
purely algebraic definition of the Hodge filtration.
The theory of MHS developed by Deligne [D2, D3] is used more and
more. The definition of the MHS includes, besides a decreasing Hodge
filtration F-, an increasing weight filtration W. Deligne showed that the
cohomology of any algebraic variety (possibly non-proper and singular)
has a natural MHS. The MHS also appears in investigations of degenera-
degenerations of algebraic varieties. Let a morphism f:X—>S define a family of
non-singular projective varieties over the punctured disk S', and let
Xo =/-'@) be the degenerate fibre. Schmid [Sm] and Steenbrink [SI]
investigated the question of what happens with the Hodge structure on
H"(Xt, Z), when t is limited to the point 0. The limit object appears to be
a MHS. The Hodge filtration of the limit MHS is in a sense the limit of the
Hodge filtration on Hn(Xt), and the weight filtration is related to the
monodromy.
The study of period maps goes back to the investigations of Abel and
Jacobi on integrals of algebraic functions. A tempestuous development of
the theory of periods of integrals begins after Griffiths's papers A968).
Griffiths studied the properties of integrals in terms of the notions of the
period matrix space and the period map which he introduced. Let us

xiv Introduction
consider a family Xt of non-singular projective varieties depending on a
parameter t € S, and defined by a proper morphism /: X —> S. Using the
connection on the fibration Я = |J tesHn(Xt), we can displace the Hodge
structures on H"(Xt) to the cohomology space H = H" (Xto) of a fixed
fibre. Considering the Hodge filtration FpHn(Xt) only, we can associate a
flag F'(t) in Я to every point t e 5,
Я = F°(t) D F\t) D ... D Fn+\t) = {0},
and hence obtain a point F'(t) of the manifold of flags of given type. We
obtain the period map Ф: S —»&~. In coordinates the period map is given
by the periods of integrals (we have to choose a basis in the homology
space consisting of continuous families of cycles, and to define the
subspaces Fp(f) by bases of differential forms). In fact the definition of Ф
is not correct, since as t moves round a loop in S, the identification of
Hn(Xto) with itself need not be the identity. So we have to consider either
a map Ф: S —> &" of the universal cover of S, or a map Ф of S to a quotient
of .^"by some group.
Griffiths found that to first order, Fp is deformed only into the subspace
Fp~l. In terms of the Gauss-Manin connection this can be interpreted as
VFP С ?2ls <8> Fp~l (the horizontality theorem). In fact, Griffiths consid-
considered polarized Hodge structures on the primitive cohomology groups
P"(Xt). He constructed the period matrix space D of all possible polarized
Hodge structures of given type, which is a submanifold of the flag manifold
distinguished by the Hodge-Riemann bilinear relations. It turns out that D
is an open homogeneous complex manifold, there is a naturally defined
properly discontinuous group Г of analytic automorphisms of D such that
M — D/T is an analytic space, and then we obtain the period map
Ф: S -> M.
The period map can be used for the description of the moduli of
algebraic varieties. Here there are problems about Torelli-type theorems.
The global Torelli problem is the question of whether the period map
Ф: Ж —> M of the moduli space of algebraic varieties of a given type is
an embedding, i.e. whether the period matrix uniquely characterizes the
polarized algebraic variety. The affirmative answer to this question in the
case of algebraic curves is the usual Torelli theorem. The local Torelli
problem is one of deciding when the Hodge structure on H*(Xt, C)
separates points in the local moduli space (Kuranishi space) of Xt. The
infinitesimal Torelli problem is one of deciding when the tangent map dФ
to the period map of the universal family is a monomorphism. The criterion
for the infinitesimal Torelli theorem to hold, which was obtained by
Introduction
xv
Griffiths A968), stimulated the appearance of many papers on this theme.
We are not able in this introduction to go in detail into the problems
touched upon above, and we refer the reader to one of the surveys on the
theory of Hodge structures and periods of integrals, e.g. to [K-Ku] and
[B-Z], where one can find other references.
The aim of this book is to transfer the ideas and notions, described
above, to the local situation - to the case of isolated singularities of
holomorphic functions/: (C+1, 0) —> (C, 0). Again we have a morphism
/: X —> S, but now the fibres are local analytic hypersurfaces in an open
set X С C+1. In Chapter I we introduce the main personage of this book -
the meromorphic Gauss-Manin connection Ж of a singularity /and the
Brieskorn lattice ^@) in it. We prove the regularity of the singularity of
the Gauss-Manin connection and the monodromy theorem. The discussion
in Chapter I is based on the classical papers of Brieskorn [Br] and
Malgrange [M]. In Chapter П we consider the limit MHS appearing on the
vanishing cohomology Hn(Xt, C) of an isolated singularity/ In the main
we follow the development as it occurred historically. Here the main
contribution comes from the papers of Steenbrink, Varchenko and Scherk.
Initially the MHS on the vanishing cohomology H"{Xt) was constructed
by Steenbrink [S3] following a suggestion of Delinge. He used an embed-
embedding of the morphism /: X —> S in a projective family Y —> S and the
limit MHS in the case of a degeneration of projective varieties [SI]. Then
Varchenko [V2, V3] proposed and accomplished the direct introduction of
the limit Hodge filtration F~ on Hn(Xao, C) (by means of asymptotics of
integrals), without using an embedding to a projective family. Following
this idea Scherk and Steenbrink [Sc-S] introduced the filtration F' in a
different way. They showed how the filtration F' is obtained from the
embedding of the Brieskorn lattice Ш^ in the meromorphic connection.
Finally, in Chapter III we consider the period map of a //-const deformation
fy(x) of isolated singularities parametrized by points у € У of a non-
singular manifold. The basis of this chapter is the papers of Saito [Sa8],
Karpishpan [Ka2] and Hertling [Hel, He2]. First we consider the period
map defined via embedding of Brieskorn lattices but not the one defined
via the limit MHS on Hn(Xt, C) as it should be according to the ideology
presented at the beginning of this introduction. Instead it is connected with
the following. Firstly, the limit MHS is determined by the embedding of
Brieskorn lattice, and, secondly, the limit MHS on H"(Xt, C) is a rougher
invariant than the embedding M^ С Ж is (because the filtration F' is
defined not by the embedding ^@) С ~Ж but by the embedding of adjoint
objects Grv3@^ С Grv^M). We give examples of the explicit calculation

XVI
Introduction
of the period maps for the deformations of unimodular singularities and
prove the horizontality theorem and the infinitesimal Torelli theorem.
Now we give a detailed description of the contents of this book. In §1 of
Chapter I we recall the definition of the Milnor fibration [Mi] /: X -> S of
an isolated singularity /: (C"+1, 0) -> (C, 0) The restriction of/over the
punctured disk S' is a smooth locally trivial fibration. We introduce the
cohomological fibration H=\J teS> H"(Xt, C) defining the topological
Gauss-Manin connection V,op on the sheaf of sections Ж —
H®6S = Rnf^Cx' ® &S1- In §2 we develop elements of the theory of
connections on locally free sheaves, the presentation of which one can find
in Deligne's book [Dl]. We pay special attention to the relation between V
and the dual to it connection V*. For Gauss-Manin connections, deter-
determined by the homological and cohomological fibrations, this leads to the
period matrix of a singularity / being a solution of system of linear
differential equations - the Picard-Fuchs equation of the singularity.
In §3 we introduce the De Rham cohomology sheaf M^R(X/S) =
J$?n(f%Q'x/s), which is a natural extension of the sheaf 3@ to the whole
disk. This sheaf establishes the connection between topology and algebra
and analysis, and reflects the analytic nature of the singularity /. §4
contains Brieskorn's calculation of the Gauss-Manin connection of the
sheaf .Ж(~2) = ^^R(X/S) in terms of differential forms. This calculation
naturally leads to the sheaf Ж^ = /*Q?/s/fl7(/*Q?^) which contains
.Ж(~2) and is also an extension of the sheaf 3@. We introduce the notion of
the meromorphic Gauss-Manin connection on the sheaf .Ж =
,^(-2) ф &s[t~1]- We also explain how to give a more conceptual descrip-
description of the calculation of the Gauss-Manin connection on Ш(~2) as a
connecting homomorphism in an exact cohomology sequence. In §5 the
main personage of this survey appears. This is the Brieskom lattice J%(°\
the third natural extension of the sheaf Ж to the whole disk S. The lattice
3%щ is defined in terms of (n + l)-forms. The identification of M@) and
J^) on S' is realized by means of the Leray derivative (the Poincare
residue), со >-> co/df = Res [co/(f - /)]. All three lattices ^~l\
i = 0, 1, 2, are the terms of an increasing filtration on Ж. The correlation
between them is contained in the diagram shown in A.5.3.4). In §6 from
Malgrange's theorem, which claims that the periods of an и-form со have
the limit limr_>o J y(t)O> — 0, we obtain a result of Sebastiani about the
absence of torsion in the sheaves Зв^1*, and hence we obtain that these
sheaves are locally free ^-modules of rank /л.
In §7 we recall the classical definition of a regular singular point of a
system of linear differential equations and Sauvage's theorem on the
Introduction
xvu
regularity of a simple singular point. We give the definition of a connection
with regular singularity and of the residue of such a connection wrt a
saturated lattice. In §8 we prove a fundamental theorem on the regularity of
the Gauss-Manin connection. We give two proofs of this important fact.
Firstly, we prove that periods of holomorphic forms give solutions of the
Picard-Fuchs equation. Then from Malgrange's theorem it follows that
horizontal sections of the Gauss-Manin connection have a moderate
growth. Secondly, using a resolution of the singularity and the sheaves of
differential forms with logarithmic poles, we construct a saturated lattice in
the meromorphic Gauss-Manin connection. In §9 from the regularity of
the Gauss-Manin connection we deduce the monodromy theorem claiming
that all eigenvalues of the monodromy are roots of unity. We give a
beautiful proof of this due to Brieskom, based on the positive solution of
the seventh Hilbert problem. The part of the monodromy theorem concern-
concerning the size of Jordan blocks is proved in (II.3.5.9). We obtain corollaries
of the monodromy theorem about the decomposition into series of the
periods of integrals of differential forms.
In §10 we consider non-isolated hypersurface singularities. Starting from
the construction of the Gauss-Manin connection as a connecting homo-
homomorphism in an exact sequence of complexes and in the spirit of [Sr], we
obtain a natural generalization of Brieskom lattices 3?M to the case of
non-isolated singularities. We give Van Straten's criterion on the absence
of torsion in the sheaves ,Ж@ and its application to the non-isolated
singularities with a one-dimensional critical set.
In Chapter II we consider the limit MHS appearing on the vanishing
cohomology H"(X,, C) of an isolated hypersurface singularity. In §1 we
recall very briefly the necessary basic definitions from the theory of MHS
developed by Deligne. In §2 we introduce the limit MHS constructed by
Schmid [Sm] for a variation (H, &') of pure Hodge structures F) on a
vector space H, parametrized by points of the punctured disk S'. The limit
Hodge filtration F- appears on the zero fibre S§ItSS of the canonical
extension 3S of the sheaf Ж' = H®3Sf to the point 0. The weight
filtration W. is the weight filtration W(N) of the nilpotent operator
N = -(\/2m) log Tu. In §3 we introduce the limit MHS on the vanishing
cohomology H = Hn(Xoo, C) (Xoo is the canonical fibre of the Milnor
fibration) according to Steenbrink. Steenbrink used an embedding of the
Milnor fibration X —* S to a projective family Y —> S. He introduced the
limit MHS on H"(YOO, C) using a resolution of singularities of the zero
fibre and the complex of relative differential forms with logarithmic poles.
We give Steenbrink's construction only very schematically, without going

XV111
Introduction
into technical details. The limit MHS on H"(XOO, C) constructed by
Steenbrink can be considered as a quotient of the MHS on Я"(Уоо, С).
From here it follows the symmetry of Hodge numbers of the MHS on
H"{Xoo, C), and also the monodromy theorem on the size of Jordan
blocks.
In §4 we consider the Hodge theory of a non-singular hypersurface Y'm a
non-singular manifold X developed by Griffiths and extended by Deligne
to the case of divisors with normal' crossings. This theory enables us to
calculate the cohomology of Yand of the complement X\ Y by means of
differential forms on X with poles on Y, and also relates the Hodge
filtration F' to the pole order filtration P\ When У has singularities, the
filtrations F' and P- do not, generally, coincide [Kal, D-Di]. We apply this
theory for the description of H"(Xt), where the hypersurface Xt С X is
the fibre of the Milnor fibration. The relative variant of this theory enables
us to obtain a natural extension S@x of the sheaf Эв to S. Then ^ is
described as the cohomology sheaf Эвх = J*$n+l(K-, d) of the bicomplex
(K", d\, d2), the terms of which are differential forms with poles, and the
degrees are defined by the order of a differential form and the order of its
pole. In the bicomplex K~ • the Poincare complex and the Koszul complex
are intertwined, and its differentials are exterior differentiation d( = d of
differential forms and exterior multiplication by the form df. The Gauss-
Manin connection and the Hodge filtration F' on S& x are defined in terms
of this bicomplex. In fact Ж'х is a Z)-module, or a differential system,
which appears in the papers of Pham [Phi, Ph2], Scherk and Steenbrink
[Sc-S]. The study of this is extended in the papers of M. Saito [Sal-SalO]
in the frames of the theory of Z)-modules and of the 'monstrous' theory of
mixed Hodge modules developed by him. This latter theory raises the level
of abstractness by at least one more order, and moves further from the soil,
in which all this began to grow. The 'new' homological algebra, derived
cathegories, perverse sheaves, the Riemann-Hilbert correspondence, etc.
are essentially used in it. Under Saito's influence the theory becomes more
and more abstract and technical. One of the aims of this book was not to
follow this line but to evolve the theory using the 'traditional' (i.e. habitual
to algebraic geometricians) language of sheaves, connections, spectral
sequences etc. and to avoid using the language of the theory of Z)-modules,
the theory of mixed Hodge modules etc. This is possible because of the
following circumstance. As is shown in §5 the operator d, is invertibli
3SX, $@x contains the Brieskorn lattice .Ж@) = FnS&x as well as
canonical lattice 2§, Jlf<0) С 3? С Mx. That is the pair J?m С 32 <
tains all the information which is of interest to us. The localization oi
Introduction
xix
differential system JM'x leads us to the classical frames of the mero-
morphic connection ^S = C&x\t)> and as is explained in the supplement
to §7, the pair ^@) с 22 happily stands localization and removes to
^M, J%?m С 32 С Ж. All this enables us to work in the classical frames
of the meromophic connection ^M.
In §6 we study the structure of the meromorphic connection Л6, its
decomposition into the sum Л& = ©a Са of root subspaces of the operator
tdt, the F-filtration on Ж. We determine the isomorphism гр:
Н^э 32'/t32 = ©_i < а^оСа between the canonical fibre of the fibration H
and the zero fibre of the canonical lattice 32 = V>~X^S, and it enables us
to introduce the MHS on Я in terms of the lattice 32.
In §7 we introduce the MHS on H = Нп{Хж, С), according to Varch-
enko [V3] and Scherk and Steenbrink [Sc-S]. At first we define the limit
Hodge filtration F' according to Scherk and Steenbrink, who transformed
the approach of Varchenko and gave the definition of F' in terms of the
differential system 5&x and the embedding ,Ж@) С 5%, and then we define
F% according to Varchenko. Such an order of exposition is connected with
the fact that, in our opinion, the constraction of Scherk and Steenbrink has
genetically a more natural motivation. In the beginning we observe the
sequence of steps leading to the constraction of F' by Scherk and
Steenbrink.
In §8 we study the main discrete invariant of a hypersurface singularity
- its spectrum Sp(/), and a more detailed invariant Spp(/) the set of
spectral pairs. To give Spp(/) is equivalent to giving Hodge numbers H^'q.
The spectrum Sp(/) is a set of (x rational numbers d\, ..., a^, where
a, = —A/2яп)Л, are logarithms of the eigenvalues Xj of the monodromy
T. It codes the relation between the semisimple part of the monodromy and
the limit Hodge filtration F\ We study the properties of spectrum. We
show that Sp(f) с (-1, и) and that the spectrum is symmetric wrt the
centre of this interval (n — I)/2. We develop the techniques for the
calculation of the spectrum. We explain: how to find the spectrum of a
(semi)quasihomogeneous singularity; how to find the spectrum in terms of
the Newton filtration defined by the Newton boundary; and in particular,
how to find the negative part of spectrum, the degree of which,
!С-1<а«о"а = Pg, is equal to the geometric genus of the singularity [S3];
how to find the spectrum of the join of isolated singularities, and in
particular, how the spectrum changes under adding squares of new
variables. This technique enables us to find spectra of all the simple, uni-
and bi-modal singularities, which we gather together in a table. Finally, we
study variations of MHS of families of hypersurface singularities. We deal

XX
Introduction
with this more widely in Chapter III, but in §8 we consider the behavior of
discrete invariants under deformations. We give the results of Varchenko
and Steenbrink on the semicontinuity of spectrum and on its stability under
ц-const deformations. At the end of §8, following Steenbrink, we define the
spectrum of non-isolated singularities and give a theorem on the relation of
the spectrum of a singularity with a one-dimensional critical set and the
spectra of isolated singularities of its Iomdin series.
In §1 of Chapter III we begin to study ц-const deformations fy{x) of
isolated singularities parametrized by the points у е У of a non-singular
variety. We explain how to glue the objects, associated earlier with an
'individual' singularity, to a family parametrized by Y. In particular, we
obtain the family of Milnor fibrations X(y), the family of cohomological
fibrations H = \J H(y), the family of meromorphic connections
~& = ®pCp, etc. In §2 we obtain the formula for differentiating wrt
parameter у geometric sections s[co](t, y), defined by a holomorphic
(n + l)-form a) = g(x, y) dx. The same formula is used for differentiating
the root components s(co, /?) of the geometric sections s[co](y) =
J2p>-is(a>, Р){у). In the case of upper diagonal deformations of a quasi-
homogeneous singularity f(x) from this formula we obtain a formula for
decomposition of s(a>, /3H0 into Taylor series in degrees of y. In §3 first
we define the period map Ф: Y —> П defined by the embedding of
Brieskorn lattices. For all Brieskorn lattices JgW(y) we have inclusions
V> -' з M@){y) D Vn~x. The period map takes a point у to the subspace
J#@H>) mod V~x in the finite-dimensional vector space V>~l/V~x, i.e.
to the point in the Grassman manifold П. Then we define the period map
Ф defined by the MHS on the vanishing cohomology. This period map
takes a point у to the Hodge filtration F(y) of the limit MHS, i.e. to the
point in the flag manifold. We give, following Hertling, explicit calcula-
calculations of the period maps of universal families of unimodular singularities.
In §4 we calculate the tangent map of the period map and prove the
honzontality theorem and the infinitesimal Torelli theorem for the period
map Ф. We compare the period maps Ф and Ф of the miniversal /i-const
deformation of a quasihomogeneous singularity. From this comparison it
follows that the Torelli theorem for the period map Ф in general is false.
Finally, in §5 we consider the 'global' Torelli problem. Varchenko intro-
introduced the notion of the Picard-Fuchs singularity PFS(f) of a singularity /
in terms of framed Picard-Fuchs equations. Hertling reformulated this in
terms of embeddings of Brieskorn lattices. We interpret it as a point in the
quotient П/Gi of the period space by the group Gx с GL(Hz) of
automorphisms commuting with the monodromy. Then the Torelli problem
Introduction
xxi
for the family fy, у € Y, concerns the injectivity of the map Ф1:
YI ~ -> П/Gi, where '—' is the equivalence relation induced by the R-
equivalence of singularities. We give Hertling's results [Hel, He2].
The reader is expected to have the knowledge and training usual in
algebraic and analytic geometry. This includes knowledge of sheaf theory
and the technique of spectral sequences.
The list of references reflects the development of singularity theory. It
includes: firstly, the original papers of E. Brieskorn, B. Malgrange, J.H.M.
Steenbrink, J. Scherk, A.N. Varchenko, F. Fham, M. Saito, Ya. Karpishpan
and С Hertling, directly connected with the considered topic; secondly,
some general papers on MHS and periods of integrals (P. Griffiths, P.
Deligne, W Schmid); thirdly, some books and surveys [Mi, AGV, Phi, Dl,
Di3, B-Z]. Some other references are also included.
Now about numbering and cross-references in this book. Each of three
chapters is divided into sections, and each section is divided into subsec-
subsections. In a subsection all claims, remarks and displayed formulae are
numerated successively in a uniform way by three numbers, the first of
which is the number of the section and the second the number of the
subsection. For example, the tag A.3.2) means claim (or formula) 2 in
subsection 3 of section 1. To refer to a claim etc. we use three numbers
within a given chapter, and four numbers, the first of which is a roman
numeral, to refer to other chapters. Thus (II.7.5.2) refers the reader to claim
2 of subsection 5 of section 7 of chapter II.
The preparation of this book was partially supported by Grant. No. 95-
01-01575 from the Russian Foundation for Fundamental Research and
Grant No. 4373 from the INTAS.

The Gauss-Manin connection
.41
1 Milnor fibration, Picard-Lefschetz monodromy transformation,
i topological Gauss-Manin connection
| /./ Milnor fibration
| A.1.1) Let /: (C+1, 0) ->(C, 0) be a germ of holomorphic function
} / = f(xo, x\, ..., х„). We assume as a rule that / has an isolated critical
; point at 0 e C+1, in other words, / has (or is) an isolated singularity. Let
В = {\x\ < e} С C"+1 be a ball of radius e, and let S = {\t\ < д} с С be a
| disk of radius д. Put X = B(~)f~l(S) and let / also denote the restriction
I of / onto X, f: X -> S. Let 5" = 5\{0} be the punctured disk,
j X, — f~\t) be the fibre over the point t € 5", and Xo =/"'(()) be a
| singular fibre, Jf' = X\Xo. Denote by /' the restriction of/ on X'
I XDX' D X,
\ /I if
| 5D5"9f.
j As is shown by Milnor [Mi], if ? and 6 <€. ? are sufficiently small, then /'
! is a smooth locally trivial fibration, the diffeomorphism type of which only
depends on the germ of/ at 0. Usually the fibration /': X' —> 'S' is called
the Milnor fibration. In the following it will be convenient for us to call the
| whole morphism /: X —> S the Milnor fibration. Any fibre Xt, t 6 5", is
t called a Milnor fibre. We can think of the singular fibre Xq as a degenera-
| tion of a family of manifolds Xt. Replacing the fibres X, by their
[ (co)homology we get a 'linearization' of the family X,.
i
I 1.2 Cohomological Milnor fibration
| A-2.1) A Milnor fibration /' defines a vector bundle H—* S' on 5", or a
locally constant sheaf (or a system of local coefficients, in different
terminology)
i
! 1

/ The Gauss-Manin connection
/65'
and it also defines a dual vector bundle
Я, = Homcr (Я, Cs.) = (J НР(Х„ С).
res'
We call Я the cohomological, and Я* the homological Milnor fibration of
a singularity /.
7.5 Topological Gauss-Manin connection
A.3.1) Denote by
Jg = Я <g> ^V
c
the locally free sheaf of sections of the fibration Я A local section w of Я
can be viewed as a family of cohomology classes w(t) depending on
parameter t. The flat system Я in the sheaf Ж allows us to translate
cohomology classes w(t) from a fibre X( to a nearby fibre and, con-
consequently, allows us to differentiate the cohomology classes wrt parameter,
and to define a connection on the sheaf (see the §2). This connection is
called the topological (or transcendental) Gauss-Manin connection and is
denoted by Atop.
1.4 Picard-Lefschetz monodromy transformation
A.4.1) A locally constant bundle Я (or Я*) defines (and is defined by) an
action of the fundamental group n\(S', t) on a fibre H, = HP(X,, C). Let
y: [0, 1] —> S' be a loop representing an element of Я]E', t). The inverse
image of the locally trivial fibration f: X' —* S' defines a locally trivial,
and consequently, trivial fibration y~lX' —>[0, 1] on the segment. A
trivialization of this family defines a diffeomorphism hy: Xt — (p~l@)
-> Xt = cp~l(l). We thus obtain the monodromy representation
A.4.2) я,E', 0 - А\йН"(Х„ С), [у] -> (A*).
Let [у] е Я]E', /) be the generator of the fundamental group represented
by a counter-clockwise oriented circle у around the origin. The linear
transformations
A.4.3) M = (hy)*: НР(Х„ С) -> Hp(Xt, С)
A.4.4) T = (h*yl: НР{Х„ С) -> Н"(Х„ С)
2 Connections 3
are called the (local) Picard-Lefschetz monodromy transformation of
homology and cohomology, respectively.
A.4.5) The Milnor fibration, the Gauss-Manin connection and the Pi-
Picard-Lefschetz monodromy are the main objects associated with a singu-
singularity / and they contain a great deal of information about this singularity.
These objects can be studied by topological methods. In the case of
isolated singularities the most important fact ([Mi]) is that the fibre X, has
the homotopy type of a bouquet S" V ¦ • ¦ V S" of ц spheres of dimension
n. In particular, the Betti numbers b, = dim H'(X,, C) = 0 for i ф 0 and n
and, consequently, H= R"f*Cx' is the only non-trivial fibration among
the fibrations Rpf*CX'- The number ju = dim Н"(Х„ С) = Ь„ (the rank
of this fibration) is called the Milnor number of the singularity /. We refer
the reader to other surveys ([AGV], ch. 1; [Di3]) for topological methods
for the study of singularities. In this survey we'll be interested in the study
of singularities by algebraic methods in which the main subject as well as
the main tool of investigation is the differential forms associated with a
singularity.
2 Connections, locally constant sheaves and systems of linear
differential equations
2.1 Connection as a covariant differentiation
Let S be a complex manifold of dimension m, and '? be a locally free sheaf
of rank n on S. The presence of a connection on <? enables us to
differentiate sections of '6 along vector fields on the base S. On the other
hand, the notion of a connection on '? is a way of having an invariant
definition of S of a system of homogeneous linear differential equations
with n unknown functions in m variables. In the following we'll be mainly
interested in the Gauss-Manin connection. First we briefly recall the
necessary facts about connections. An excellent exposition of this subject
can found in Deligne's paper [Dl].
B.1.1) Definition A connection on a quasicoherent sheaf of ^-modules
(S is a C-linear homomorphism
V: $ -> Q^ ® 8' = Qls(g)
Г- s
satisfying the Leibniz identity
= dg <8i s+ gVs,

/ The Gauss-Manin connection
B.3.5)
k=\
where the holomorphic functions Г,*(х) are called the connection coeffi-
coefficients (wrt the basis e and coordinates x).
Denote by 9, = 9/9x, the differentiation wrt the coordinate xt, and let
V,- = Vg.: ? —> ? be the covariant derivative wrt the vector field dj. Then
B.3.4) turns into a system of equalities
B.3.6)
7=1
" i = 1, ..., n, к = 1, ..., m
B.3.7) Definition A local section s of the sheaf Й" is called horizontal if
Vs = 0, i.e. s € KerV, V: ? —> Q^ <g> r6.
The condition of horizontality of section s can be written in the form
VkS = 0, к = 0, ..., m, i.e. in the form of system of homogeneous linear
differential equations of the first order
B.3.8)
This is a system of nm equations with n unknown functions yi{x) in m
variables x\, ..., xm.
If dim S = m — 1, Hs a coordinate on S, &>,;,¦ = r,y(f)ctt, and Г = (Г,у@)
is the matrix of connection coefficients, then the operator D = dt acts on a
section s = ey according to the formula:
B.3.9) Dy = dy/dt + T(t)y
and the condition for horizontality of 5 is written in the form of a system of
ordinary differential equations
B.3.10) dy/dt = -T(t)y.
2.4 The integrable connections. The De Rham complex
B.4.1) A connection V = Vo: '& —> Q.\ <8> % can be extended to a C-linear
homomorphism of sheaves
f' s f-s
by means of the equalities
V,(uj О s) - dcu (Э s + (-1)'u> Л V0(s).
B.4.2) The C-linear homomorphism
2 Connections
Д = Vi о Vo: # -> ?
is called the curvature of the connection V on <?, so that R is a section of
the sheaf .Уёгт^б, Q|(?T)) ~ Q^S{'S^/с {'<€)). A connection V is called
integrable if R = ViV0 = 0.
One can show that for vector fields X\, X2 ? 0s and a section s ? & one
has
R(Y, JMM — Vv Vv V — Vv V v ? — Vrv vl?
and the condition of integrability of connection V is equivalent to the
condition 4[x,Y] — [Vx, Vy] for arbitrary vector fields X and Y. If
dimS=l, then any connection is integrable. A local calculation of
curvature R shows that the condition of integrability of V is equivalent to
the classical condition of integrability of the corresponding system of
linear differential equations.
B.4.3) One can show that V,-+] V,(ai <g> s) = со Л R(s) (the Ricci identity).
Therefore the condition of integrability of a connection V, R = 0, is
equivalent to the condition that
B.4.4) Qs(%): 0 -» &s О Z ^Qls О %^-* ¦ ¦ ¦ "^ Q™ О $ -> 0
is a complex, i.e. V,+iV,- = 0. The complex Q,'s{rf) is called the De Rham
complex with values in a locally free sheaf % with an integrable connec-
connection V.
2.5 Local systems and integrable connections
B.5.1) Let S be a connected and locally connected topological space. A
locally irec sheaf of vector spaces E on S, i.e. a sheaf locally isomorphic to
a constant sheaf C", is called a local system on S.
B.5.2) The fundamental group H\(S, xq), xo ? S, acts on the fibre
Ещ — С". The functor E \—»?^ establishes an equivalence of the category
of local systems on S with the category of complex-valued representations
of the group 7i\{S, xo).
B.5.3) Now let E be a local system on a complex manifold S. Consider a
locally free sheaf %> of holomorphic sections of E, К = fi s ®c E. Then
there is a canonical connection V on % for which the sheaf of horizontal
sections coincides with E, Ker V = E. Here V is defined by the formula
V(gs) = dg s

10
/ The Gauss-Manin connection
We have (a>j, y,)' = 0 = (Daij, y,-) + (coj, D*yt) = Tv + Г*. ¦
B.6.6) Let«i, ..., u>u be a basis of local sections of the sheaf J@, and let
у be a section of Ж*. Then the functions
/,@ = (coby>, ...,/„(/) = <<»„, Y)
can be considered as coordinates of у in the basis dual to {со,}:
In particular, if у is a section of #*, i.e. a horizontal section of J^ , then
V*y = 0 and the coordinates of у satisfy the system B.3.10) of differential
equations of the connection V*. Thus if у is a section of #*, then the
functions I\(t), ..., Ift(i) satisfy a system of differential equations
B.6.7) | : |=Г'@| ! |,
Л
where Г = (Г,,) is the matrix of coefficients of V in basis a)\, ..., <oA.
If у\,...,Уц form a basis of sections of H*, then the columns
Q; = (Q^j(t), ..., Qw(/))', Qy = («,, yj), form a basis of solutions, and
the matrix
B.6.8) Q(i) = (Qij(t))
is the fundamental matrix of solutions of the system /' = Г'(/)/.
3 De Rham cohomology
3.1 The Poincare lemma
We return now to our main subject of investigation - the Milnor fibration
/: X —» S of a singularity of holomorphic function / and the connection
on the locally free sheaf Ж = Rpf*Cx' ® &s' on S', defined (topologi-
cally) by the local system Rpfjf<Cxi- In order to pass to differential forms
we have to use the Poincare lemma.
C.1.1) The Poincare lemma (in its holomorphic version) asserts that if X
is a complex manifold, then the De Rham complex (Qx, d) is a resolution
of the constant sheaf Сл-, i.e. the sequence 0 —> Cx —> 6'x —»Qx —> ... is
exact.
This implies that for a complex manifold X we have an isomorphism
HP(X, C) ~ MP(QX), where UP(QX) is the hypercohomology of the De
Rham complex. If, moreover, .AT is a Stein manifold, then (see below for the
relative case) the cohomology HP(X, C) is isomorphic to the De Rham
3 De Rham cohomology 11
cohomology, i.e. to the cohomology of the complex (H°(X, Qx, d) (closed
forms/exact forms on X). Turning to the relative case, we have the
following generalization:
3.2 Relative De Rham cohomology
C.2.1) Definition The sheaves J^^R(X/S) of hypercohomology of the
De Rham complex of relative differential forms of a morphism /: X —> S
C.2.2)
Qx/S: 0 ¦
wrt the functor of direct image /* are called the relative De Rham
cohomology
C.2.3) ^^{X/S) = Upf^Qxls).
Here Qpx/S = QP = Qpx/f*Qxs Л Qf".
The main tools for working with hypercohomology are the first and the
second spectral sequences of hypercohomology, which for the complex
Q'x/s are of the form
'ЕГ =
np+"MQx/s),
C.2.4')
C.2.4")
Here J&p(-) denotes the cohomology sheaves of a complex of sheaves,
and Л«/*О>/5 is the complex ... -+ Л«/*О^ -> R4f*®x/S ~*
3.3 De Rham cohomology for smooth Stein morphisms
C.3.1) A connection of the relative De Rham cohomology with topology
is established by the relative Poincare lemma:
Proposition If a morphism /: X -*S is smooth, then
C.3.2) Sg
Proof The relative Poincare lemma asserts that if / is smooth, then there
is an exact sequence 0-+f-l<9s -»• ?2>/s, i.e. the De Rham complex
QX;S is a resolution of the sheaf f~x&s- Hence Wf^(Qx,s) =
^p/(/1^>)- It can be shown that the canonical homomorphism

12
/ The Gauss-Martin connection
is an isomorphism. И
C.3.3) Corollary If /: X -* S is the Milnor fibration, then the sheaf of
relative De Rham cohomology J^0K(X/S) is a natural extension to the
whole S of the sheaf 3V = Rpf*Cr <8>cs. &? on S',
This follows from the smoothness of the morphism /': X' —» S'.
C.3.4) Proposition For the Milnor fibration /: X —» S we have
i.e. the sheaves of De Rham cohomology are the cohomology sheaves of
the direct image of the relative De Rham complex.
Proof The morphism / is Stein and hence the first spectral sequence
C.2.4') degenerates 'Ep'q = Rif*(Qx/s) = 0 for q s* 1 and hence
•f 3?Qp) WM&) ' "
3.4 Coherence theorem
This is one of the fundamental technical results of the theory.
Brieskorn [Br] proved that if /: X -* S is the Milnor fibration of an
isolated singularity, then the sheaves JtZg^X/S) are coherent sheaves of
^-modules.
For proper morphisms coherence follows from Grauert's famous direct
image theorem. Brieskorn obtained the proof of the coherence by embed-
embedding the morphism /: X —> S in a projective morphism /: Y -+ S. Later
generalizations and other proofs of the coherentness theorem were ob-
obtained by Hamm [H], Buchweitz-Greuel [B-G] and van Straten [Sr].
3.5 On the absence of torsion in the De Rham cohomology sheaves
C.5.1) The next important question is one about the absence of torsion in
the sheaves J@gR(X/S). We'll discuss this in §6 after introducing the
Gauss—Manin connection on these sheaves. For an isolated singularity
/: (C"+1, 0) -> (C, 0), the absence of torsion in $fgR(X/S) forp = n
was proved by Sebastiani [Se]. For p < n, torsion is absent by virtue of the
3 De Rham cohomology 13
presence of a connection on these sheaves. From this, C.3.3) and A.4.5) it
follows that for an isolated singularity we have
{
0
for p = 0,
for 1 *? p < n,
for p = n.
The absence of torsion in the sheaves J&'m(X/S) is also important
because in such cases the Milnor number, pM = dim H'(X,, C) = bt (the
i th Betti number of fibre X,), is given by the formula
and the germ of the De Rham cohomology sheaf yff'm(X/SH can be
calculated 'upstairs' as shown in the following proposition.
3.6 Relation between У@р{/*пу) andf*J%fp(Qy)
C.6.1) Proposition If /: X-л S is the Milnor fibration of an isolated
singularity / (or, in the more general case, of a concentrated non-isolated
singularity, see §10), then the De Rham cohomology sheaf
Jf enters into the exact sequence
C.6.2)
where the last sheaf is concentrated at the point t0 = 0 € S, and the two
first sheaves are locally free of rank ц(р) = bp{X,) on 5", and at the point
to = 0 the first of them has a zero fibre and
C.6.3) Jtf</tQy)h =
where (X, xq) = (C+1, 0).
Proof Consider the second spectral sequence of hypercohomology C.2.4")
• =
y)) =>
As /' is smooth, the sheaves $?i(Qy) are concentrated on Xo =/-'@)
for q э= l and hence ИЕ{Л = 0 for q э= I and p s* I. For q = 0 we have
and "Epfi =

14
/ The Gauss-Manin connection
This involves the exact sequence C.7.2). The germ Лр/*(/ l&s)ta =
Hp(Xo, &s,t0) = 0 is zero because Xo is homeomorphic to a cone over the
point x0 and hence is contractible. ¦
Thus we obtain that for an isolated singularity / the Milnor number is
equal
C.6.4)
(because .3$
the result of Sebastiani has no torsion).
4 Gauss-Manin connection on relative De Rham cohomology
Let /: X ->• S be the Milnor fibration of a singularity /: (C+1, C)
—> (C, 0). We have local systems - cohomological fibrations H =
RPf'*Cx- = U Ks'Hp(Xt, C) on S' = S\{0}, and hence there are corre-
corresponding connections on Jff = H ®cs. &*$• which we call topological.
On the other hand we have an identification of 3% with the De Rham
cohomology (§3), J& ~ $?z>K(X'/S') and a natural extension of this sheaf
over the whole S, .%fgR(X/S) ~ JtSp<J*Gxls). We want to calculate the
connection on 3&gK(X'/S') corresponding to the topological connection
on Je and to extend it to M^{X/S)
4 Gauss-Manin connection on relative De Rham cohomology 15
4.1 Identification of sheaves of sections of cohomological fibration and
of relative De Rham cohomology
D.1.1) The identification of .M with .3fgR(X'/S') is obtained by means
of integration of differential forms (De Rham's theorem).
Let Я* = U teS-Hp(X't, C) = HomCj. (Я, Cs>) be the homological fi-
fibration on 5". This is a local system dual to H. We have a non-degenerate
pairing
X
, where
= H*
с?
Consider a local section a of the fibration H*. We can think of a as a
continuous family of cycles o(t) € Hp(Xt, C) on hypersurfaces X,, which
is obtained geometrically by continuous deformation of a cycle o(t0) on
one of the fibres Xh. The homology classes of cycles o(t) do not depend
on the deformation (if о is a section over an open set lying in some sector
with the center at the origin 0 € S).
Now let со be a local section of the sheaf .%j?,r(X'/S'). The cohomology
class со is presented by a cocycle, i.e. a p-fovm со such that d<y = d/ Л rj.
Consider the function I(t) which is the integral of the form со over the
cycles of the family a, i.e. the value of I(f) at a point t is equal to the
integral of the p-form a>\x, over the /?-cycle o(i) С X,
D.1.2)
-f
J<7
-.
We obtain a non-degenerative pairing
D.1.3) Jff^jiX'lS1) X Зё* -» ^y, (o>, a)
which identifies J!?gR(X'/S') with the sheaf dual to M*, i.e. with the
sheaf 3f.
D.1.4) To verify that /@ is, in fact, a section of the sheaf <9$'> i-e. that
/@ is a holomorphic function, and also to calculate /'@> we need Leray's
residue theorem. This theorem and its application are natural general-
generalizations of the integral Cauchy formula g(z0) = (l/2jri)Jg(z)dz/
(z — z0) and its application for the proof of the existence of g'(z) and
decomposition of a holomorphic function into series in the standard course
of complex analysis.
A fibre Xt С X is a submanifold of complex codimension 1 and hence
of real codimension 2. There is defined the coboundary Leray operator
D.1.5) d: Hp(Xt) -+ Hp+1(X\Xt).
One needs to consider a tube neighbourhood N of the submanifold Xt in X

16
/ The Gauss-Manin connection
(fibered on disks). Then the boundary dN С X\X, is a fibration on circles
over X, and д assigns to a cycle in Hp(X,) its pre-image on dN wrt the
projection dN —> X,.
D.1.6) Lemy 's residue theorem asserts that
/@
Ja@ 2OT
dfAco
E<7@ f —
This theorem is called the residue theorem because co\x, is the Poincare
residue of the form d/ Л co/(f - t), co\Xl = Res*, [d/ Л co/(f — t)]. In-
Indeed, we can consider the equation у = /(*) - t = 0, of a smooth fibre X,
as a local coordinate on X. Then dy-df (t = const). However, the
Poincare residue is defined in the following way: we- 'chip' a multiplier
dy/y = df jf — t and restrict it to the subvariety Xt: у = 0.
D.1.7) Theorem The function I(t) = \O(t)<» is holomorphic on S".
To prove this, we apply the integral presentation of I(t) in D.1.6). If t
varies in a small neighbourhood of a fixed point t0, we can assume that
Eст@ = Ect are the same for different t. The form d/ Aco/[f(x)- t]
depends holomorphically on t and we can apply the standard theorem of
analysis asserting that an integral of a form depending holomorphically on
a parameter t over a fixed chain also depends holomorphically on t. The
same presentation enables us to calculate /'(?) as the derivation of an
integral depending on a parameter.
4.2 Calculation of the connection on a relative De Rham cohomology
sheaf
D.2.1) We proceed to the calculation of the connection on 3@gR(X'/S').
Both this connection and the topological connection on M are dual to the
connection on 38* = H* ®cy<^V. defined by the homological fibration
Я*. By B.6.3), if D = Va/dr is the covariant derivative of the connection
on ?fgR(X'/S') and o(t) is a local section of if, then {со, о)'
= {Deo, o). We will calculate the derivation {со, о)' of the function I(t) =
I<o,o(t) = {со, о). We use the integral presentation D.1.6) and differentiate
the integral wrt the parameter t
4 Gauss-Manin connection on relative De Rham cohomology 17
/'(o=-f --d (l f <*/лвЛ_i f dfAa>
6oU)f{x) - t
Thus {Dco, a) = {rj, o) for any section о and hence Dco = 77.
D.2.2) We obtain the following rule for calculation of the connection on
Jt?gR(X'/S'), which one can (try to) carry word for word to the De Rham
cohomology sheaves M^K{XjS) to obtain the Gauss-Manin connection
defined by Brieskorn [Br]:
V:
or
Wd/dl = D:
Take a p-fona со representing a cohomology class in .>$qR(X/S). As this
is a cycle in the complex Q'x/s> we have dco — 0 in ?2^, i.e.
dco = df А т], where г] е Qx. Then
D.2.3) D(co) = a class of the form r\ or
V(cw) = d/Л D(cy) = df Ar/.
4.3 The division lemma. The connections on the sheaves $f?R
for p ^ n — 1
We have not yet obtained a connection on the sheaf .3@gR(X/S) because
we have not yet varified that the form г] represents a cohomology class in
3ggR(X/S), i.e. that it is a cocycle in /*Q>/S, i.e. that drj = df A ? is
divisible by df, i.e. dr] is a coboundary in the Koszul complex (see
A0.5.1)). It is only clear that dt] is a cocycle in the Koszul complex, i.e.
that d/ Л drj = 0. Indeed, we have dco = df A rj, hence d(dcw) = —df
Adri = 0.
D.3.1) The so-called division lemma is the assertion about the vanishing
of the cohomology of the Koszul complex. Recall the definition.
Let A be a commutative ring and g\, ..., gm € A. The cohomological
Koszul complex K'(g) is the complex

18
/ The Gauss-Manin connection
О _> А -» Лт -» Л2Лт -»...-
in which the differentials are homomorphisms, ЛМт ^-> Лр+1Лт, which
send <pto gAcp,g = gxe\ + ... + gmem ?Am - ®Т=]Ае>.
In our context the Koszul complex appears in the form of the complex of
differential forms. Let ХЪе a complex manifold of dimension n + 1, and
f(x) be a holomorphic function on X. Consider the complex (Qxis, df):
D.3.2)
d/Л,
Qpx
d/Л
оГ1
0
in which the differential is the exterior product by the 1-form df. The
cohomology of this complex
is called the Koszul cohomology.
It is obvious that for a point x ? X the complex ?i'XtX is the Koszul
complex, in which A = 6}x^ = «^o+'.o. A>n = &x,x ~ ®"=o^x,x&Xi> and
the sequence g is the sequence of partial derivatives f'^, ...,
/*„> d/ = f'xq dxo + ... + f'Xn dxn. There is a well-known lemma:
D.3.3) Division lemma (De Rham) If g\, ..., gm is a regular sequence,
then the cohomologies of the Koszul complex are
Hp(K-(g))=l°' P<J^
i.e. if to € ЛЛ4 m and p < m, then
gA со = О <$Ф со = g/\ rj, for some rj € Л^'Л. ¦
D.3.4) If 0 € X is an isolated critical point of a function /, i.e. 0 is a
unique solution of the system f'^ = ... = f'Xn = 0, then f'^, ..., f'Xn is a
regular sequence, and the division lemma asserts that Жр(/) = О for
p < n, and the sheaf ,%n+x(J) = Qnx+ x/df Л Q"x = Q?^ is concentrated
at the point 0 and Qnx/ls ~ <9x,o/Jf, where J/ = (/^, ..., f'xJ is the
Jacobi ideal of/ The algebra
D-3.5) Qf = <9Xfi/Jf
is called the local ring of the critical point 0 of / [AGV] or the Milnor
algebra or the Jacobi algebra of the singularity / The gy-module
Q "x+}s ~ Qf is often denoted by
D.3.6) Qxfs = Q/ or Qf.
4 Gauss-Manin connection on relative De Rham cohomology 19
Thus if/ is an isolated singularity, then we have the exact sequence
0 —> @x —-> Q^. _>...—> Q^. —+ QJ+1 —> Qf —+ 0;
and if u> € й? and p *? «, then df Лео = 0 ¦& <x) = df Arj.
D.3.7) We now return to the definition of the connection on J%?R(X/S).
We saw that the (p + l)-form drj is a cocycle, df Adrj = 0. However, if/
has an isolated singularity and p+ 1 =s n, i.e. /? < n — 1, then drj is a
coboundary as well, drj = df A ?. This means that the form 7 defines a
cocycle in the complex f*Q'x/s and we indeed obtain a connection
D.3.8) It is easy to verify the correctness of the definition of D = Vd/dt,
i.e. that the class rj = D(co) is independent of the choice of the form со in
a class of .M^K{X/S). Moreover, D really is a covariant derivation, i.e.
it satisfies the Leibniz identity. We have: dco = df Л rj, rj = D(co);
d(gft>) = dg Л со + g-dco = d/ Л (dg/df)co + d/ Л grj, i.e. D(gco) =
g'co + gD(co). (Here /is a local coordinate on S and dg = (dg/df) df.)
4.4 The sheaf •Jg = /*Q?/s/d(/"*Q$)
The previous construction does not work for p = n, because in this case
the (n + l)-form drj need not be a coboundary, i.e. it may not be divisible
by d/ We obtain that the operator D = Vd/rf, = dt acts from .
not to itself but to a greater module,
D.4.1) V: Ж
The sheaves J^r
0
„_,.
and 'yg are defined from the complex
* dii—1 n d,.
f * у Ж XfS J ^ л. I о J *
1 = Kerdn/Imdn_i, '^f = Cokerdn_i = Дй^
Hence there is the exact sequence
d _i_i
D.4.2) 0 —> ^5dr(a/o) —> .Ж —* J if^xlS ~* '
and since the sheaf f*Q"x+/\ — /*(Q/) is concentrated at the point 0 6 5,
the restrictions of J^f^R(X/S) and '.%! coincide on 5'.
D.4.3) We obtained a connection in some more general sense [M]: let
E С F be two (9 = ?f'^-modules; then a C-linear map D: E —> F,

20
/ The Gauss-Manin connection
satisfying the Leibniz rule D(ge) = (dg/dt)e + gD(e), e e E, g € @, is
called a connection in the pair (E, F). Using localization, we obtain the
connection in the old sense.
4.5 Meromorphic connections
Let б = @s$ = C{f} and let Ж = 6(t) = 6\rx\ be a field of germs of
functions meromorphic at the point 0 € S. Here Ж is a field of fractions
of the ring &. Let M = Же\ ф ... Ф Ж^е^ be a vector space over Ж,
dim^ M = ц. The definition of a connection transfers word for word to
this situation.
D.5.1) Definition A meromorphic connection on M is a C-linear map
V: M —» Qs,0 ®^ Mora C-linear map
D = Vd/rf/: M^M,
satisfying the Leibniz rule: D(grn) — g'm + gD{m).
D.5.2) As in §2 a meromorphic connection D is defined by the matrix
T(t) = (Г/Д0) of connection form coefficients in a basis e\, ..., ец of a
vector space M, D{ej) — 5^f=ir,y(f)e(-. The condition of horizontality
s = ey € Ker Z) is given by a system of linear differential equations
y' = —T(f)y with a singularity at the point 0 € 5, in which coefficients
are meromorphic functions.
D.5.3) Now if V is a connection on an 6>-module E, then it is naturally
extended to a connection on the vector space M = E(t) = E ®^ Ж by the
'rule of differentiation of a product (fraction)'
s
D.5.4) In the same way we can localize the situation in the case of a
connection Don a pair (E, F), where E с F and F/E is torsion, i.e.
dime F/E < oo. In this case E ®r Ж = F ®f Ж = M and the connec-
connection D: E —» F defines a meromorphic connection on M.
D.5.5) Definition A lattice in a vector space M, dim^- M = ц, is an @-
submodule E с M of rank /г, i.e. a finitely generated submodule E such
= M.
4 Gauss-Manin connection on relative De Rham cohomology 21
D.5.6) To return to our discussion: let /: S' С S be an inclusion and
Ж = ЯФсу <^s' ^ .^dr(^75') be a locally free sheaf on 5'. Denote by
Ж[0] С г* (Ж) a subsheaf of the direct image of Ж, in which the germ at
0 e S coincides with the localization of the module .^R(I/5H as well as
'Жо. We obtain a meromorphic Gauss-Manin connection V on the sheaf
Ж[0], in which the locally free submodules Ж^Х/Б) С 'Ж С ЖЩ
are lattices (we will prove in §6 that there is no torsion in the sheaves
and 'Ж and hence they are locally free).
4.6 The Gauss-Manin connection as a connecting homomorphism
D.6.1) We can give a more conceptual definition of the Gauss-Manin
connection V: Ж^Х/S) -^>Qlx<Sirs '¦%¦ Denote by Qx/S a 'shortened'
complex Q'x/S' m which the term Qnx+}s is changed by zero:
v: 0
&X/S
^x/s ~~* ®x/s ~~* 0-
Then .%ff(f*?lx/s) = 3&p(f*?lx/s) =
for p « л - 1, and
х/8
By the definition of the sheaves of relative differentials there is an exact
sequence of complexes
7=T.-1 4>
S
0, where
d/ Л
If/ has an isolated singularity, then the division lemma means that q> is a
monomorphism, i.e. the sequences
D.6.2)
со) = d/ Л to = 0, then ш = df Arj, i.e. ш = 0
are exact. Indeed, i
in ?2^, i.e. Ker <p = 0.
Applying the hypercohomology functor IR/,,, to the exact sequence of
complexes, or in our situation of Stein morphism /, applying functor /* to
the sequences D.6.2), we again obtain exact sequences and hence an exact
sequence of complexes
D.6.3)
0
or in more detail:

22
0 -
i
f x—>
II
fx —,
I
0
0
i
Qi®/*''
I
ftQx
i
i
0
/ Tfte Gauss-Manin
0
i
*\ "
—>...—> /*QJ,j —>...—»
I
0
connection
0
i* ^/s
/,QJ —»
i
/* ^J/s ~*
I
0
0
1
i*
i
0.
t
—> 0
-.0
-.0
5 Brieskorn lattices
23
Consider the connecting homomorphism д in the exact cohomology
sequence of this exact sequence of complexes
D.6.4)
Remembering the rule of calculation of the connecting homomorphism д
in an exact cohomology sequence, we obtain that
d: &&l(X/S) -> Qls® Jf^X/S) or d: .
&l(X/S) -> Qls® Jf^X/S) or
(if we identify Q^ = 6>s&t with <^',y) exactly coincides with the Brieskorn
definition of the Gauss-Manin connection V, i.e. it coincides with the
operator d, = Vd/dt: .3fgR(X/S) -> .%?gR(X/S). For p = n we obtain
accordingly the Gauss-Manin connection д = V,
д: J^^iX/S) -*Q\2> 'Jg.
f' s
Now note that X is smooth and by the Poincare lemma the complex Q^
is a resolution of the sheaf Cx and hence .%fp(f*Qx) = RPf*(Q-x)
= Rpf*Cx- However, this sheaf has a zero germ at the point 0 6 S by
C.7.1). From D.6.4) we obtain:
D.6.5) Corollary The Gauss-Manin connection on the De Rham co-
cohomology gives the isomorphisms
d,:
0).
(and 0 - С - J^m(X/SH
D.6.6) Remark As noted by Malgrange [M], this enables us to calculate
the Betti numbers bp = dim Hp(Xt, C) of the Milnor fibre (i.e. the Milnor
numbers) purely analytically without appealing to a result of Milnor about
the homotopy equivalence of X, to a bouquet S" V ... V S" of spheres.
The presence of a connection on J^^R(X/S) for 1 =? p ^ n — 1 involves
(see F.1.1)) that .Ж^Х/S) are locally free sheaves of rank bp. Moreover,
Ker<9, is a local system of rank bp (by the Cauchy theorem on solutions of
a system of linear differential equations). But dt is an isomorphism and
hence Ker<9, = 0 and bp = 0 for 1 =s p ^ n — 1.
D.6.7) We give one more proof of the coincidence of the Gauss-Manin
connection on JtfpR(X'/S') (defined by Brieskorn) with the topological
connection on Ж = Rnf*Cx' ® @s'-
To prove that under an isomorphism of sheaves the connections corre-
correspond one to another, it is sufficient to show that horizontal sections of
each of them correspond one to another. Again consider the exact
hypercohomology sequence D.6.4) corresponding to the sequence of
complexes D.6.3) on S', i.e. for/': X' -> S',
Hence Ker V = Kerд = Im (Ки/*(й>) -> R"/*(Q>./SO- Complexes
Q^-. and Q'x'/s' are resolutions of the sheaves Cx' and f~ld?s', respec-
respectively. Hence Ker V corresponds to
Cs
i.e. to horizontal sections of the topological connection.
S Brieskorn lattices
First we summarise briefly the results of §4. Let /: X —» S be the Milnor
fibration of an isolated singularity and H = Л"/*Сл" = (J KX'H"(Xt, C)
be the cohomological fibration on S'. The locally free sheaf
Jg = H ® (9S' on S' is identified with the De Rham cohomology sheaf
.$@pR(X'/S'), to which we transfer the topological connection on .Ж. The
sheaf .M^R{X'/S') naturally extends to the locally free sheaf J%pR(X/S)
on the whole of S. The attempt to extend the connection V onto
JtfnR(X/S) leads us to a locally free sheaf 'j^f D :Ж?К(Х/Б) coinciding
with J%fpR(X/S) on S'. The covariant differentiation gives an isomorphism
d, = Vd/dr: JefgR(AT/5)^'J8f. The identification of the sheaf
<M = f*Qx/s/d(f*Q'x~/ls) as well as of the sheaf ^^{X/S) with the
sheaf 3% on S' is realized by means of the De Rham theorem: a local

24
/ The Gauss-Manin connection
section of 'Зё is represented by a class of an и-form со on X, the
restrictions (O\x, (these are forms of the largest degree on X,) give a family
of cohomology classes w(t) € H"(Xt, C), i.e. a section w = s[co] of the
sheaf 36 = R"f*CS' ® cfS: The germs of the sheaves 36?R(X/SH and
'36q at the point 0 ? S have the same localization wrt t, which is a germ
of the sheaf 36[0] с /*(.Ж). The connection V defines a meromorphic
connection on 3&[0], and 3$qK{X/S) c '-^ c 3?Щ are two lattices in
36[0].
5.1 Brieskorn lattice 6
E.1.1) Brieskorn considered one more natural lattice, 6 in 36[0],
associated with (n + l)-forms for which the identification with 36 on S' is
realized by means of a Poincare residue.
The division lemma implies the exact sequence
Q
Л+1
X/S
0.
Taking the quotient by the subsheaf dQ"X/s С QX/s an(* applying /*, we
obtain an exact sequence
E.1.2)
where the sheaf
E.1.3) 6 = f
is called the Brieskorn lattice.
^
0 -4 '36^ 6-^f*Qf -* 0,
E.1.4) The sheaf /*Q/ = f*Qx+/s is concentrated at the point 0 € S and
so the sheaves '.96 cd/A "j^ coincide on S'. The sheaf "Jgf is free of
torsion (§6) and so 6 С 36[0].
E.1.5) The identification of ".^^ with 36 or with 'J#|S- is realized in
the following way. The inclusion '36 с % in E.1.2) is given by multi-
multiplication by d/, hence the inverse mapping must consist of division by d/.
This is possible on X' because the Koszul complex is exact there. Namely,
let со be a (n + l)-form on X representing a section of ".M. The fibres Xt
are hypersurfaces defined by equations f(x) — t = 0, t = const. For t ф 0
the fibres X, are smooth and we can consider the function f(x) -(as a
local parameter у — f(x) — t; dy = d/ because t = const. The form со of
the largest degree can be represented in the form со = d/ Л гр. Then the
5 Brieskorn lattices
25
restriction ip\Xi does not depend on the form гр and it is nothing but the
Poincare residue of the form co/[f(x) — t],
CO
f-f
We obtain a family of forms on fibres X,, i.e. a relative и-form (more
exactly a section ofQx,,s,/dQx~L,) which is denoted by
E.1.6)
CO „ CO
— = Res
df f-t
and is called the Gelfand-Leray form of the form со. Hence we obtain a
section w = s[co] of the sheaf Jtf, where w(f) € Hn{Xt, C) is represented
by the form V|jr, = Res^-, co/[f{x) — t]. The section s[a>] Varchenko
[AGV] calls a geometric section of the sheaf j^f. Thus we obtained the
identification "J^fp- = ^.
5.2 Calculation of the Gauss—Manin connection V on ' .9ё
E.2.1) We defined the covariant derivative D = dt = Vd/dt of the Gauss-
Manin connection on J/fpK(X/S), dt: .M^R(X/S) -> '36, which to an n-
form со representing a section of Зё^уЯ{Х/S) assigns an и-form dt(co)
defined by the rale
dco = df A dt(co).
Taking into accoun/the inclusion 3if^R(X/S) с "Зё, we see that the
covariant derivative dt: J^f^R(X/S) —» 6 reduces simply to the differen-
differential d, d,(co) = dco. The same formula defines the extension of the connec-
connection on 36^K(X/S) to '3?,
E.2.2)
5,: 'Зё -»• 'Ж д,(со) = dco.
5.3 Increasing filtration on
E.3.1) We join together the exact sequences D.4.2) and E.1.2) connecting
the sheaves .%?pR(X/S), '3f, ".Ж and operators d,. However, before this
we introduce new notation for these sheaves following that used by K.
Saito [SK] (the reason for this new notation will become clear below).
These sheaves are first in a series of sheaves forming an increasing
filtration.

/ The Gauss-Manin connection
26
Put
E.3.2)
All these sheaves naturally coincide (are identified) with the sheaf 3$ on
S", and their germs at the point 0 € S can be calculated on X by means of
complexes of germs at the point x = 0 e C"+!:
= H\Q-x/SiX),
E.3.3)
E.3.4) Proposition There is a commutative diagram
0^ j?f(-i)dZ>A Л
ltd, = d
QX~J + dQ
x~J
where Q/ = Qx+}s, and r is a mapping to the quotient. Both operators dt
in the diagram are isomorphisms.
Proof The commutativity of the diagram is obvious. Let us verify that
d,: 3@(~X) —> J3?(O) is an isomorphism. Analogically we can verify that the
second operator д,: ,Ж() —> .Ж(~1} is an isomorphism and, moreover,
besides this was already verified in D.6.5). It is useful to carry out such
reasoning by means of the following commutative diagram
E.3.5) 0
T
T-d/
T-d/
fQ"
o,
T
T
T
which we will call a 'fragment' because it is a part of a double complex, in
which the complexes (f^Qx, d), and (f^.Qx, —d/Л) are bound and
whose natural origin will be explained in (II.5.3.10).
Obviously, dt: J^' —> .Л^0) is an epimorphism because an element of
.Ж@) is represented by an (n + l)-form a> on X and it can be represented
in the form a> = drj because the complex (Q^, d) is exact in dimensions
5 Brieskorn lattices
27
5= 1. Let us show that dt is a monomorphism. Let ш € f*Qx represent an
element of J^' and let d,([(o]) = 0 in J??@), i.e. dw = d/Ad?7,
t] G /*Q?"'. We have to prove that [со] = О in ^(~1), i.e.
<u = d/A?+d?i, ?, and ^le/^QJ. We have dco = d/Arf?7 =
—d(df Л rj), i.e. d(co + d/ A 77) = 0. Again from the exactness of (Q^, d)
we have that w + d/ A rj — d^i, i.e. ш = —d/ A 77 + d?i. ¦
It follows from Proposition E.3.4) that we can restore
of dfA: J&-V с .^@) anda,: J&~1) ^M°\
E.3.6) .Ml~2) = {co€ M(~X): d,(w)
This enables us to define an increasing filtration on
E.3.7) j?<-« = {« g J^(-*+1): d,(w) g
2^ by means
We obtain a commutative diagram
E.3.8) ... с .у^(-к) с ... с
)dc:A
с J?f(-1)dc:Aig?@)-:>/*Q/ -»о
ц
0,
... С Ж(-к~1) С ... С ^-9> С ^(-2> С f
where, as it is easy to see, all inclusions have the same quotients,
E.3.9)
0
С ^
0.
5.4 A practical method of calculation of the Gauss-Manin connection
E.4.1) Let V,//,/, = dt: ^S —> ^M be a meromorphic Gauss-Manin con-
connection on a vector space ^M = ^@) <8>^s,0 ^ over the field
^ = <fs,o['-1] D.5.6). To calculate the connection dt practically it is
convenient to use the Brieskorn lattice 3^ = Qx^/df Л dQ^1 because
Q x+q is a locally free module of rank 1 and differential forms w = g(x) dx,
dx = dxb A ... A dxn, are identified with functions,
$ ~ ^,0, 01 = g(x) dx
Consider the exact sequence E.1.2)
0
d/Л
С
f
f
in
Under these epimorphisms the module Q? =
with the local Artinian algebra g/ = @
D.4.6) is identified

28 / The Gauss-Manin connection
As we know E.2.2), the connection d, on the lattice Л?<-~1) С Лву"' is
calculated by the formula:
E.4.2) for d/ Л ? 6 d/ Л M~X) we have d,(df A ?) = d|.
On the other hand, the ^o-module /*Q/ is concentrated at the point
0 e S. Hence there exist such jfc that /к.Ж{0) с ^(~u. Hence to calculate
9, for an element ш € .Ж@), we need first to 'drive' w into Зё(~х^ and then
to use the previous rule: if / ku> = d/ Л ?, then d,(fka)) = dg; but
9,(/ *a>) = kfk~]a> +f kd,u> and hence we have in ^Ж
E.4.3) d,a> = --<y + ^|.
^ / /*
The (^5,o-module .Жт is generated by a class of the form dx. So to 'drive'
.M@) into M{~X), it is sufficient to find such A: that
E.4.4) /*dx = d/A?,
where ? = JXo^'OOdxo Л ... Л dx, Л ... Л dxn € QJ0 is an и-form.
However, &f f\% = Y^i=0(—X)lhi(x)f'Xlux and so the calculation of the
connection of J^@) is reduced to the question of finding к such that
E-4.5) fk&(f'xo>--->f'xJ = Jf
and explicit expression of/* by the derivatives f'x. (the calculation of the
form |). The identification of QJ^1 with 6>x,o, oj = g(x) dx <-> g(x),
enables us to rewrite the calculation of the connection dt in terms of
functions g{x) € б x,o.
Let us consider an important example.
5.5 Calculation of the Gauss-Manin connection of quasihomogeneous
isolated singularities
E.5.1) Let /(x) = J^mcmxm e C[xo,..., xn] be a quasihomogeneous
singularity of degree d = 1 with weights wt(x,) = w,, where for
m = (mo, ..., mo), xm denotes x™0 x™«. In this case the connection
d, on .Ж@) is calculated explicitly and completely. This arises from two
circumstances. Firstly / € J, i.e. к = 1, and from the Euler identity
YLi=owixif'xi = f(x) f°r tne form ^ such mat /W dx =. d/ Л | we have an
explicit formula
E.5.2)
Л ... Л dx,- Л ... Л дх„.
(=0
Secondly the image of the inclusion M^~X) с -%fm coincides with
fj&<® = t.%f(°\ i.e. with a submodule of ,Ж@) generated by the maximal
ideal (t) с ^^o- So (by Nakayama's lemma) to find a basis a>\,..., w^ of
5 Brieskom lattices
29
the ^5,o-module .^@), it is sufficient to find a basis in a vector space
/*Qr '=QT~Qf= &xfi/Jf.
Let xm = x™° ... x™" € g/ be a monomial basis in g/, m e A,\A\= [i.
Then the forms <um = xm dx, m ? A, represent a basis of the i^^o-module
J%{0\ From /(x)dx = d/A| we have f(x)com =d/Axm| and this
implies that
1
/0
Putting
E.5.3)
we then obtain:
[-co
-ш
;=o
con
i=0
E.5.4) Proposition For a quasihomogeneous singularity f{x\ deg/(x)
= 1, wt(x,) = wt, the monomial basis a>m € Ж(№>, m € A, \A\ = /л, is a
basis of eigenvectors of the operator /9,,
гд,а>ш = [а(?и) - l]com. ¦
Thus in the case of a quasihomogeneous singularity the Brieskom lattice
J^?@) is saturated, and the residue of the connection Resj^mV is a
semisimple operator (see §8).
E.5.5) Corollary The monodromy Tof a quasihomogeneous singularity is
semisimple and its eigenvalues are
E.5.6) Remark In general the Brieskom lattice J^@) is not saturated, i.e.
in its basis the system of differential equations defining horizontal sections
has a pole of order > 1. Scherk [Sc4] developed methods of construction of
the saturation j^@) of the lattice :Ш®\ the choice of a convenient basis in
j^(°) and explicit calculation of the connection dt. The calculations are
rather non-trivial. Scherk carries out his method in detail for the singularity
x5+y5+x2yz in [Scl]. This is a singularity of type Тг,5,5 in the
classification of Arnold [AGV]. It is equivalent to the singularity
(x3 + y2)(x2 + у3). This was the first singularity whose monodromy has
infinite order to be discovered [A]. For the results of the calculation for the
singularities Tp^r (see (II.7.3.5) and for the singularity ax5 + y6 +xAy,
a € C, see [Sc-S]).

30 / The Gauss-Martin connection
6 Absence of torsion in sheaves .M^'^ of isolated singularities
/: X —* S be the Milnor fibration of an isolated singularity
fibration of an isolated
"') be the sheaves introduced in E.3.7).
= /*Q"/'/d/ Л d(/*
Let /: X —* S be the Milnor
/: (C+1, C) -+ (C, 0), and let .^
In particular ^() = .Щ^Х/S
the Brieskorn lattice. These are coherent ds-modules on a disk S. We'll
prove that these sheaves are torsion free (this is a result of Sebastiani - see
[Se, M]) and it involves that they are locally free sheaves of rank /л.
6.1 The presence of a connection implies the absence of torsion
F.1.1) Lemma Let ? be a finitely generated (^-module, & = @o$. If
there is a connection on E, then E is free.
Proof Assume for simplicity that r = 1. Let e\, ..., ep be a minimal
system of generators of E. To construct this one can consider a basis
ёь ..., ~ep of a vector space E/mE and the pre-images e, € E of elements
в/. Then by Nakayama's lemma e\, ..., ep generate E. We next prove
that e\, ..., ep are free generators, i.e. there are no relations between
them. The proof is by reductio ad absurdum. Assume that
f\e\ + ... +fpep = 0 is a non-trivial relation, fi(t)E(9. Then
/i@) = ... = fp@) = 0 because otherwise the system of generators is not
minimal. Let к be the minimal order of vanishing of functions /,@. which
can appear in such relations. Then k> I. Let us take the relation
J2fiei — 0 with minimal к and apply the operator D = Vd/d, to it. We
obtain a relation between the e, for which at least one of the coefficients
has the order of vanishing =? к - 1, and this contradicts the minimality of
к. Indeed, if D(ej) = ?Г=1Гуе<> then
If/}0 has order к at 0, then /H as well as the entire coefficient has order
6 Absence of torsion in sheaves M< l) of isolated singularities 31
F.1.2) Lemma A.1) implies that the sheaves J^^K{X/S) are locally free
for p «? n - 1, because there are Gauss-Manin connections on them.
Before passing to the sheaves .Ж(~1) we prove an important result.
6.2 A theorem of Malgrange
F.2.1) Theorem [Ml] Let со be an n-form on X (a section of the sheaf
^(~1)), and let у be a section of the homology fibration Д* (a family of n-
cycles) in a sector containing the zero ray, arg t = 0. Then
lim
co = 0
Proof Consider a preimage Y — f
the axis arg t = 0. Let Г be
Y(t0) € Я"(*,о, С).
!([°. А)]) С X of a segment [0, t0] on
an и-cycle on X,o of a class
0 f f0
Sinee Xo is contractible, 7 is also contractible and H"(Y, C) = 0. There-
Therefore Г = дА, where Д is an (n + l)-chain on Y. Then by Stokes-Herrera
theorem
I(to) = co= \ со = \ dco.
Jy(<0) Jr Ja
Consider an (и + l)-chain Д, = /"'([0, ty DA,t6 @, t0]. Then
/(?)=[ со = [ rfco.
Jy(O Ja,
Indeed, Д = Д, + Д', where Д' is a chain on /"'([*, fo]) and
9Д' = Г —Г,. Hence Г, is a cycle representing y(t) and /(fo) = /дско
= JAf dco + jA. dco = JAf dw + Jrw - Jr,o> = Jд, dw + 7(r0) - 7@- It
follows that 7@ = /д, dco. Now by a classical theorem of analysis
lim,_o J д, dco = J д0 dco, where До = Xo П Д is an и-chain on JSTq- How-

32
ever, Xo\Sin;
Hence /До dco = 0.
/ The Gauss-Manin connection
is an и-manifold and dco is an (n -I- l)-coboundary.
6.3 Connection on a pair (E, F)
F.3.1) We make one more general remark. Let E с F Ъе ?f-modules,
& = &s,o, and let D be a connection on the pair (E, F), i.e. D: E -* F is
a C-linear homomorphism satisfying the Leibniz rule. Denote by E* and
Fx the torsion submodules of ? and F. Then
F.3.2) D(ET) С FT.
Indeed, assume that e e P and tke = 0, then 0 = D{tke) - ktk~le
+ tkD(e). Multiplying this equality by t, we obtain tk+lD(e) = 0, i.e.
D(e) € F\
F.3.3) Going to quotients E = E/E1 and F = F/Fz, we obtain a connec-
connection D on the pair (E, F). Moreover, if dime F/E < oo, i.e. F/E is a
torsion, then D defines a meromorphic connection on the space
?\ where Ж is a field of fractions of &.
6.4 Sheaves.^-p) are locally free
F.4.1) Theorem .^f<-2> = .i?fDR(X/5), ,i?f<-!> ^QJ/rfC/^QJ-1) and
j^(°) = /+QJ+!/d/ Л dC/^Q^) are locally free ^-modules of rank p.
Proof It is sufficient to prove that these sheaves are torsion free. As
J^f(-2> с Jg(~X) С Jtf@) are submodules it is sufficient to prove that Jgf@)
is torsion free. Let J^~l) = E, .^0) = F. We then have E с F and
Z): ?^ F is an isomorphism by E.3.4). Assume that the torsion module
FT^0. Then ETCFT and from F.1.1) and F.3.2) it follows that
E* ф Fx otherwise there is a connection on ET = FT and hence Fx = 0.
As D: E~c-> F is an isomorphism, there exists cufi E1 such that
D((o) e FT. Here со is represented by a section of .Ж'' such that
<У|5' t^ 0 and D(w)|5' = dco\S' = 0. Let y(f) be a local section of the
homological fibration. Then (d/dt) /j,(()«y = /У(,) da; = 0. This means that
the function I(t) = $y(t)(O = const. However, by Theorem B.1)
lim,^o I(t) = 0. Hence I(t) = 0 for any section y(t) of the fibration Я*.
This means that со is zero as a section of the sheaf .^f = ^1'
со e ET.We therefore obtain a contradiction and hence FT = 0.
i.e.
7 Singular points of systems of linear differential equations 33
7 Singular points of systems of linear differential equations
7.7 Differential equations of Fuchsian type
We begin by recalling the notion of a regular singular point of a linear
differential equation of order fj. or of a system of linear differential
equations of first order.
G.1.1) In the 1860s L. Fuchs A833-1902) investigated linear differential
equations
«<"> +
+... + pM-i(t)u' + Pll(t)u = 0,
in which pi(t) are meromorphic functions on the complex plane (or on
some of its domain) having a finite number of singular points. We are
interested in a local case and so we'll assume that pt(t) are holomorphic on
a punctured disk S' = {0 < \t\ <6} and consider a singular point t = 0.
By Cauchy's theorem, in a neighbourhood of a point t0 € S' the equation
has a fundamental system of solutions u\(t), ..., ицA), which can be
continued along a counter-clockwise oriented circle around the origin.
Returning to the initial point we get a new fundamental system of solutions
which is obtained from the initial system by multiplication by a mono-
dromy matrix T.
G.1.2) Fuchs proved (see below) that if Ab ...,A* are roots of the
characteristic equation \T —XE\ = 0 of multiplicities m\, ..., m^, ctj =
(l/2jri)logA,, then we can choose a fundamental system of solutions in
the form of linear combinations of functions
taJ(pfl(t), f logt-cpj2(t), ...,ta>log' t-cpjmj, j=\,...,k,
where (Pji(t) are holomorphic on 5".
G.1.3) A singular point t = 0 for which functions (pj\(i) have only poles
(but not essentially singularities) was later called a regular singular point.
G.1.4) Fuchs proved A866) that for a singular point t = 0 to be regular it
is necessary and sufficient to have aj(t) = pj(t)tJ holomorphic at 0, i.e. to
have pj(t) of the form pj{i) = aj(t)/tJ. If all dj(t) = a, = const, then we
obtain the so-called Euler equation. Linear differential equations in which
all singular points are regular, are called equations of Fuchsian type.

34 / The Gauss—Manin connection
7.2 Systems of linear differential equations and connections
Consider a system of linear differential equations
G.2.1) I :
[Ум = Ъ
or
G.2.2)
У = At)y,
where у = {y\, ..., ум)' is a column of unknown functions, A{f) = (<zy@)
is a matrix of system, and <z,y(f) are functions holomorphic on the
punctured disk S'.
By Cauchy's theorem on the existence and uniqueness of solutions in the
neighborhood of any point t0 € S' there exists a fundamental system
(basis) of solutions Ji J^, where ~y~j — {y\}- Ум])'- Writing solu-
solutions 7y as columns of a matrix Y(f) = (yij(t)), we obtain a fundamental
matrix of solutions. Going around the origin 0 € S in a counter-clockwise
direction we obtain a linear transformation in the space of solutions of
system G.2.2) in the neighborhood of the point to. This is the monodromy
operator T. Let Г also denote the monodromy matrix wrt basis y~\, ..., J^,
i.e. columns of T consist of coordinates of the vectors Tyj. Then after a
complete revolution around the point 0 E S the matrix transforms to
Y(t)T =&,..., у„)Т.
In other words a space of solutions of the system y' = A{f)y defines a
local system E on S'. The local system E defines a connection V on the
trivial sheaf g1' =E®r-y<9s- ^ &%¦ = @%x6»vej. If T(t) = (Г,у@) is the
matrix of connection V coefficients wrt the basis e, then we can consider
the system G.2.2) as a system of differential equations denning horizontal
sections of the connection if we set
G.2.3) A(t) = -T(t).
Then the monodromy of system G.2.2) is nothing but the monodromy of
the connection V.
G.2.4) The matrix A{i) of system G.2.2) (the connection matrix) depends
on the choice of a basis in <%. If Q(t) is a transition matrix to a new basis
and new coordinate finctions v are connected with old coordinate func-
functions by у = Q(i)v, then a substitution in the equation y' = A{f)y shows
that this equation transforms to v' = B(t)v, where В = Q~XAQ - Q~x Q.
7 Singular points of systems of linear differential equations 35
7.3 Decomposition of a fundamental matrix Y(t)
The matrix function Y{i) has a singularity at the point 0 e S. As in the case
of functions the singularity is characterized by: (i) a revolution around the
point 0, i.e. by ramification; and (ii) approaching 0 e S. We can separate
these effects.
Let Г be a monodromy matrix of system G.2.2) wrt a basis J\, ..., j^.
Let us represent the invertible matrix Г by the form
G.3.2) T=zbl{R.
Recall that using series we can define functions f(A) in matrix A. In
particular, for a matrix R € М{ц, С) we can consider a matrix function
x,(R\nt)k
G.3.3)
= gR\n, =
k=0
II
The function tR has the same monodromy as Y(t): after rounding 0 € S
the function tR is multiplied by e2mR = T. Hence we can represent Y(t) by
the form
G.3.4) 7@ = Z{t)-tR,
where Z{f) is a single-valued analytic function on S'.
The matrix Г and hence the matrix R depend on the choice of the basis
Уи ¦¦ ¦ ,Jfi, i.e. on the choice of the fundamental matrix Y(t). Changing
Y(f) to Y(f)C, where С € GL(n, C) (transition to a new basis), we obtain
c~lRC
Y(t)C = Z(t)C(C-ltRQ = Z(t)C-t
i.e. the matrix R transforms to a matrix C~lRC. Hence changing the basis
У\> ¦¦¦ ,~Уц, we can assume that R has a Jordan normal form. In particular,
if R is a Jordan block of order к with eigenvalue a,
G-3.5) „,
la \
1 a 0
= aE + N, where N =
1
1 a
then
/ 1
G.3.6) tR = e*ln/ = ta-
(lnf)
n-1
V(n-l)! ¦"
,0 ... 1 0,
\
0
lnr 1

36
/ The Gauss-Manin connection
T=
where A = е2л:1а is an eigenvalue of the monodromy, and Tn is an unipotent
part of the monodromy,
G.3.7) N = — logTu.
7.4 Regular singular points
G.4.1) Definition A singular point 0 € S of system G.2.2) is called
regular, if in the representation of the fundamental matrix У(/) = Z(t)tR
the matrix function Z(/) is meromorphic, i.e. functions z,y(/) have only
poles (and haven't essential singularities).
G.4.2) This definition can be reformulated in terms of the rate of growth
of solutions as / —> 0. Let g(i) be a multivalued function on S'. Let us
make a cut along the real ray U+ П S'. Let Si be S' with this cut. One says
that a function g(t) has a moderate rate of growth at 0, if any branch of
g{f) on S\ grows at most as a degree of \/\t\, i.e. |g(/)| ^ Л/|/|" for some
и. A vector function has a moderate rate of growth, if all its coordinates
have a moderate rate of growth.
G.4.3) It is easy to see that a singular point 0 € S of system G.2.2) is
regular if and only if any of its solutions has a moderate rate of growth
atO.
We have characterized regular singular points in terms of solutions of
the equation. Now we'll do it in terms of the equation G.2.2) itself.
7.5 Simple singular points
G.5.1) Definition The point / = 0 is called a simple singular point of
system G.2.2), if its matrix A(t) has at the point r=0a pole of at most
first order, i.e. if A(t) can be written in the form A(i) = A0/t + Ai(t),
where Aq is constant and A\(t) is a holomorphic matrix. The matrix Aq is
called a residue of system G.2.2) at the point 0.
A system G.2.2) with a simple singular point is written in the form
G.5.2) tdly = [A0+tAl(t)]y,
where d, = d/dt.
G.5.3) Proposition If t = 0 is a regular singular point of system G.2.2),
7 Singular points of systems of linear differential equations
37
there exists a meromorphic change of unknown functions у = Q(t)v (Q(t)
is a meromorphic matrix) which transforms system G.2.2) to a system with
a simple singular point.
Indeed, let Y(t) = Z(t)tR, where Z(t) is a meromorphic matrix. Let us
take Q{f) = Z{t), i.e. make the change у = Z(t)v. As Y = Z-tR is a
solution of the system y' = Ay G.2.2), we obtain that this change trans-
transforms system G.2.2) to the system
v' =-R-v,
with a simple singular point. Moreover, we obtain a system of special
(Eulerian) type, for which A\(t) = 0. Obviously, these singularities are
regular.
G.5.4) Lemma A system of linear differential equations
У =-У>
G-5.5)
where R € М(/л, С) has a fundamental matrix of solutions
Y(t) = tR
and, consequently, has a regular singularity at the point 0 € S.
The proof of this consists of an explicit solution of system G.5.5) by
'school' methods. Namely, a system of linear equations y' = Ry with
constant coefficients and in variable z has a fundamental matrix of
solutions Y(z) = e& (it is a 'differential equation with separable vari-
variables'). A change of the variable z = In / in system G.5.5) leads to a
system y' =Ry. Hence system G.5.5) has a fundamental matrix of
solutions Y = Vn' =df tR.
A linear change of unknown functions у = Су in system G.5.5) leads to
a transform of the matrix R, R i—> C~XRC. By such a change we can
transform R into a Jordan normal form. Thus, system G.5.5) falls apart
into several disjoint systems corresponding to Jordan blocks. Denote by
<%a'q a system G.5.5) where R is one Jordan block of order q with
eigenvalue a:
{&«'<>) о" = Ry,
where R = aE + N is a triangle matrix in G.3.5).
7.6 Simple singular points are regular
The classical Sauvage theorem (Sauvage, 1853-1920) asserts:

38
/ The Gauss-Manin connection
G.6.1) Theorem (Sauvage, 1886) If / = 0 is a simple singular point of
system G.2.2), then it is regular.
This theorem follows from the following theorem, a proof of which can
be found in the [C-L]:
G.6.2) Key-Lemma (Lemme-clef) If the residue Ao of an equation
ty' = [Ao + tA\(i)]y with a simple singular point has no eigenvalues,
which differ by an integer, then there exists a holomorphic change of
unknown functions Q(t)y, which transforms this equation to an equation
G.5.5) (i.e. reduces it to the Eulerian case).
The condition about eigenvalues in this lemma can be satisfied by means
of the Lemma about shift G.7.6). We'll prove this below.
G.6.3) Thus, a singular point 0 € S of a system y' = A(t)y is regular if
and only if there exists a meromorphic change у = Q(t)v, transforming the
system to a system with a simple singular point.
G.6.4) Remark Moreover, by the classical Horn theorem (Horn, 1892)
such a matrix Q(t) can always be found in the form of a polynomial
Qit) = Qo + a / + ...+ QNtN, Qt G M{n, C).
J. Moser and D. Kutz (see [Br]) obtained an estimate for the degree N
expressed in the order of pole it of matrix A(t). Namely, let
A(t) = t~k(^2°l0Ajt') be a meromorphic matrix with a pole of order к > 1
and a nilpotent matrix AQ of rank r. Then we have:
G.6.5) Proposition A system y' = A(t)y with a regular singular point can
be transformed into a system with a simple singular point by means of a
change у = Q(t)v, where Q(t) is a matrix of polynomials of degree
N « 2[(k - 2)(л - 1) + r].
A practical method of finding the matrix Q(t) is reduced to a solution of
a system of algebraic equations. Unknowns of this system are the elements
of matrices Qo, ¦¦¦, Qn- Equations of this system express the condition
that after the change у = Q(t)v we get a system with simple singularity,
i.e. the matrix QTX AQ — Q~x Q' has at most a pole of the first order.
7.7 Connections with regular singularities
Let 3G = @\t~x\ be a field of fractions of the ring <9 = <9s,o,
M = ф;=1.Же, be a vector space over Ж, and V: M -> Qlsfi <g> M be a
7 Singular points of systems of linear differential equations 39
meromorphic connection on M D.6.1). If a basis e in M is chosen, and
Г = (Г,у(/)) is a matrix of coefficients of connection forms in this basis,
then the horizontality condition V(w) = 0 of a section m = ey is written in
the form of a system of linear equations G.2.2), where A(i) = — Г(/).
G.7.1) A connection V is called regular at the point 0 € S (or 0 ? S is
called a regular singularity of a connection V), if the corresponding system
G.2.2) has a regular singularity at 0. Taking into account the relation
between regular and simple singular points and the fact that Г(/) is the
matrix of operator dt = ^d/dt in basis e, we obtain the following reformu-
reformulation:
G.7.2) Definition A lattice IcMis called saturated, if it is stable under
the operator tdt, i.e. td,(L) С L. A meromorphic connection V on M is
called regular, if there exists a saturated lattice in M.
Indeed, if a lattice L is stable under tdt, then the matrix of the operator
dt, i.e. Г(/), has at most a pole of the first order. The converse follows from
a commutator relation
G.7.3) [tdt, g] = tg' for g G &.
Indeed, if in a basis e\,..., e^ of a lattice L the matrix Г(/) has at most a
pole of the first order, then for base elements td,(ej) € L. Then for
€ L we have
tgfa € L,
i.e. L is stable under tdt. Again from a commutator relation [td,, t] = t it
follows that for a saturated lattice L its ^-submodule tL is stable under
tdt.
G.7.4) Definition The residue of a meromorphic connection V on M wrt
a saturated lattice ? is the endomorphism
Res0 V = td,\ L/tL -* L/tL
of the vector space L = L/tL, dime L = /г.
If a connection V has a matrix of connection coefficients
Г@ = T0/t + T\(t) in a basis of L, where Ti@ is a holomorphic matrix,
then ResoV has the matrix Го in the corresponding basis.
We can change the eigenvalues of the transformation Го by adding
integers by means of a change of L. This is based on the commutator rule

40
/ The Gauss-Martin connection
[d,, t] = \, i.e. dtt= td,+ \. It is straightforward to verify that the
following commutator rule is satisfied
G.7.5) [td, - {a + \)]r-t = t{td, - a)r.
From this we obtain:
G.7.6) Lemma about shift Let L be a saturated lattice and let a be an
eigenvalue of the linear transformation Reso V = td, on the space
L — L/tL. Consider a root subspace E in L corresponding to a. Namely, if
r is the multiplicity of a (the multiplicity of a as a root of characteristic
equation), then there is a partial Jordan decomposition
X = Ker{td, - a)r © Im(id~, — a)r = 1 © F.
Let L = E © F be a decomposition of L, which gives this decomposition
mod {tL). Then a lattice
U = F © F, where ?" = ??
is also saturated. The eigenvalues of Reso V wrt V i.e. the eigenvalues of
Reso V: L'/tL' —> V/tL' are the same as wrt Z, except the eigenvalue a.
And a for L becomes a + 1 for L'.
Analogously, if we change E to E' = (l/f)?> then a is changed to
a-1.
Prao/ Let g G ? С L. This means that (fd, - a)rg G fZ,, i.e.
(fd, - a)rg= t-h, h ? L.
Consider the element tg G E' = tE. Then from G.7.5) it follows that
{td, -{a + l))r{tg) = *-(rd, - a)rg = *2A.
This means that zg G Ker(*d, - (a+ l))rmod(*2Z,), i.e. mod (*?')• Thus
subspace tE = E' gives a subspace in L'/tL' corresponding to an eigen-
eigenvalue a + 1 of the residue Reso V on the lattice L'. Ш
G.7.7) To summarise: if V is a meromorphic connection on a vector space
M over Ж, then there exists a basis e in M such that M = ®Ma'4, where
Ma'q denotes a subspace, generated by q basis vectors, with respect to
which the matrix -Г is Jordan with the eigenvalue a. If we wish, we can
change a by adding integers. For example, we can assume that all a belong
to the interval — 1 < a =s 0.
7.8 Residue and limit monodromy
Let % = (B^&s'tj be a free sheaf on S and V be a connection on ?T|y.
The monodromy operator Г is a linear transformation on the fibre
8 Regularity of the Gauss—Manin connection
41
= %, ® @s,t/"*- Properly speaking, it depends on the point t,
T=T{t).
G.8.1) Proposition If V is a conneetion meromorphie at the point 0, i.e. it
is a connection on the sheaf &[0] = % ®r 6\_\/t\ and if t = 0 is a simple
singular point, then the monodromy transformation T ean be extended to
the point 0, Blim^o T{t) = To, and To is eonneeted with Res0 V by the
relation
la —e
Proof In a basis e\, ..., e^ the matrix of eonneetion eoeffieients T{t) has
the form T{t) = T0/t + Ti{t), where Го is eonstant, and Ti{t) is a holo-
morphic matrix, and Tq = Reso V. Horizontal seetions s = ey satisfy a
system of differential equations y'=—T{t)y, or dy = — T{t)ydt. To
calculate the monodromy we restrict this equation to a eirele. Let {p, cp)
be polar coordinates on the complex plane t = pe"p. Then
dt = ip el<p dcp + Q19 dp, and on the circle p = po = eonst we have
dt = ip el(p dcp. Substituting for t and dt to the equation for dy we obtain
dy = — {T0/peiq> + Ti)yipeil?> dcp, i.e. on the eirele the differential equa-
equation has the form
dcp
The monodromy transformation Tat a point (p0, <Po) is obtained if we take
the value of a solution Y of this equation at the point {po, cpo + 2л),
satisfying the initial condition У\(ро,щ) = E, where E is a unit matrix. We
see that the differential equation depends continuously on p and cp, and
there exists Hir^o T = To, where To is obtained from the equation with-
p = 0, i.e. dy/dcp = -iToy. It follows that Y = е41»1? and Го =
у — _-2тГ0 т
2\(р=2я — e • ¦
G.8.2) Remark Properly speaking, it is wrong to say that To is eonjugated
with T{t) for t near to 0. However, if a eharaeteristie polynomial of the
monodromy T is independent of t, then it ean be ealeulated by means of
Го = Res0 V.
8 Regularity of the Gauss-Manin connection
Let /: X —> S be the Milnor fibration of an isolated singularity
/: (C+1, C) -+ (C, 0), and let Vd/d, = d,: J& -> Ж be the meromorphie

42 / The Gauss-Manin connection
Gauss-Manin connection on the sheaf Ж — Жт ®< s &s[t~l]> a restric-
restriction of which to S' coincides with Ж = Rnf*Cx <8>cS' &sl- A fibre at the
point 0 € S is a vector space Жо = Ж\' ®r s0 Ж of dimension (x over
the field Ж - ?.{t}[t~x], see D.6.6). A fundamental result about the
Gauss-Manin connection (Ж, dt) is its regularity. There is a filtration on
Ж consisting of lattices Ж(~^ E.3.7), where
To prove the regularity of the connection one can either construct a
saturated lattice L in ./M G.7.2) or show that in some basis any horizontal
section has a moderate growth G.4.3). We'll use both methods.
8.1 The period matrix and the Picard-Fuchs equation
Let Ж* = Я* ®Cs, @s, it = U t€S-Hn(Xt, C), be a sheaf dual to Ж,
and let V* be a connection dual to the Gauss-Manin connection V on Ж.
Horizontal sections of the sheaf Ж are families of cycles
y(t) € Hn(Xt, C), obtained by spreading a cycle in one fibre to nearby
fibres.
(8.1.1) Definition Let w be an и-form on X or a section of the sheaf
Ж(~х^ represented by this form, and let y(t) be a horizontal (multivalued)
section of the sheaf Ж*. A (multivalued) function on 5"
W =
is called a period of the form <w.
i = CO
)y(t)
(8.1.2) This name originates from the following classical situation. Let со
be a closed 1 -form on a manifold X. Let us consider a multivalued function
F(t) — J^ со, the integral of со over a path connecting a fixed point to with
a point t. Then the values of F(t) at the point t differ by periods, i.e.
integrals of со over closed paths, i.e. over 1-cycles y.
(8.1.3) Let ct)i, ..., Wf, be a basis of (local) sections of the sheaf 3&
represented by и-forms on X. If classes of these forms generate the germ at
0 of the sheaf Ж(~1) (or Ж(~1)), then coi ш^ is a basis of sections in
a neighborhood of the point OeS. Let T(i) = (Г,у(ф be a matrix of the
Gauss-Manin connection coefficients wrt this basis B.3.5). Then the
horizontality condition of a section о in coordinates y\, ..., Уц wrt a basis
Regularity of the Gauss-Manin connection
43
o)\ Wft is written in the form of a system of linear differential
equations B.3.10)
/ = -T(t)y.
In a dual to the coi, ..., 0^ basis of sections w*, ..., w* of the sheaf Ж*
the matrix of the connection V* is equal to Г* = -Г' by B.6.5). By
B.6.7) we obtain:
(8.1.4) Proposition If y(t) is a horizontal section of the sheaf Ж*, then
the period integrals
/i@= f o>i I/,(t)= (ом
Jy(O Jy@
are solutions of the system
(8.1.5) y' = T\t)y.
This system is called the Picard-Fuchs equation of the Gauss-Manin
connection in the basis u>\, ..., (Of,.
Indeed, Ij(t) IM(t) are coordinates of y(t) in the basis со*, ..., со*
and system (8.1.5) expresses a condition of horizontality of a section y(t)
of the sheaf Ж*.
(8.1.6) If у 1, ..., уM is a basis of horizontal (multivalued) sections of the
sheaf Ж*, then the period matrix
Q@ = (QjKO), where Щ = [ a,, \^i,j^n,
hi
is a fundamental matrix of solutions of the Picard-Fuchs equation
(8.1.7) It is easy to see that the matrix (Q')~'@ is a fundamental matrix
of solutions of the system y' = -T(t)y expressing a condition of horizon-
horizontality of a section of the sheaf Ж in basis co\, ..., coM.
8.2 The regularity theorem follows from Malgrange's theorem
(8.2.1) Regularity theorem Iff: (C+1, C) -> (C, 0) is an isolated singu-
singularity, then the singular Gauss-Manin connection V on Ж^К(Х/Б) has a
regular singular point at 0 € S.
(8.2.2) Originally the regularity theorem was proved by P. A. Griffiths
A968) in the global case, when/: Y -* S is a projective morphism. Recall

44
/ The Gauss-Martin connection
that the regularity theorem means that periods of integrals of the family
/: Y —> S have a polynomial growth 1/1*1^ at the point 0, and Griffiths
obtained an estimate of the integrals' growth. Brieskorn [Br] obtained the
desired estimate of integrals' growth in a local situation from the corre-
corresponding Griffiths's result about meromorphic differential forms on fa-
families of algebraic varieties.
(8.2.3) From (8.1.7) and from Malgrange's theorem F.2.1) asserting that
for any period I(t) = J7(,)u>, lim,^o ДО = О, it follows that horizontal
sections of the Gauss-Manin connection have a moderate growth. This
proves the regularity theorem (8.2.1).
(8.2.4) Afterwards other proofs of the regularity theorem were obtained,
some using a resolution of the singularity and others not. There are two
proofs in [M]. The second of these is based on Malgrange's criterion for
the regularity of a singularity, using the index of the operator dt- We'll
sketch one more proof of the regularity theorem (P. Deligne, see also [H]),
using a resolution of the singularity and logarithmic differentials.
8.3 The regularity theorem and connections with logarithmic poles
(8.3.1) The Gauss-Manin connection is a meromorphic connection V on
the sheaf
where 6's[Q] = &s[t~x], and Va/d/ = d, is a connecting homomorphism
in the hypercohomology exact sequence D.7.4). Now let /: X —> 5 be a
morphism but not necessarily the Milnor fibration. When / is Stein, we
have
To prove the regularity of the connection V we have to find a saturated
lattice in 3f[0] G.7.2).
(8.3.2) Denote by QX/S[XO] = Qx/S ®/-^s f~lS>s[0] a complex of
relative meromorphic differential forms on X with poles at Xo = f~l@).
We have an isomorphism
8 Regularity of the Gauss-Manin connection
The sequences of complexes
45
Q
x/S
0
is exact on X', because the restriction /to X' = X\X0, /': X' —» 5", is a
smooth morphism. Hence the kernel Ker(d/A) is concentrated on Xo and
is killed by ®y-i^,. /~'<^ s[0]. We obtain an exact sequence of complexes
0 -» Q-x/s[X0] -» QX[XO] -» QX/S[XO) -» 0
and the Gauss-Manin connection dt = Vd/dr is a connecting homomorph-
homomorphism
V: R"/*Q>
in the exact hypercohomology sequence corresponding to this exact
sequence.
(8.3.3) Consider a resolution of the singularity of the morphism/
X0CX Л XDX0
7\l/
S
i.e. a proper morphism лг: X —¦ X such that л: X\Xo^*X\Xq, where
Xq — jTx (Xo) and (Xo)red is a divisor with normal crossings.
Let QX(XO) = Q^(logXo) С Q|[X0] be sheaves of differential forms
with logarithmic poles, and let
Qpx/S(xo) =
?
be the corresponding locally free sheaves of relative differential forms.
Locally we have f = z1?, ..., zvkk and df/f = ?*=iv/dz;/*; e QX(XO)
can be completed to be a basis of QX(XO). Hence the complex (QX(XO),
(df/f) A is exact. We have an exact sequence of complexes
0
QX(XO) -+ Q-X/S(XO) -» 0
(8.3.4) Furthermore, the sheaves U'f^Qx.s{Xo) are coherent ^j-
ules and there are isomorphisms
®
Thus we can change the sheaf ЗёЩ =
U"f*Q-x/s[X0].
"f*Qx/s[X0] to the sheaf

46
/ The Gauss-Manin connection
(8.3.5) We assert that a sheaf L = Unf*Qx/s(X0)/Tors is the desired
saturated lattice in M"f*Qx ,S[XO], i-e. it is stable wrt the operator tdt.
(8.3.6) Indeed, we have a commutative diagram
Д /
l-j Г Г
о -> Щ}8[х<л й^ &к[х0] -> ®x/s[Xo] -> о
where the columns are exact sequences of complexes. We obtain a
homomorphism of the exact hypercohomology sequence of the upper line
to the analogous exact sequence of the low line. A fragment containing
connecting homomorphism is
It
1<p
We have that Im cp is the desired lattice L because this diagram shows that
д,Щ С (l/f)I. ¦
(8.3.7) Remark Arguing as above Hamm [H] proved the regularity of
the Gauss-Manin connection on sheaves R'f'*Cx' ®cs¦ @s' —
Rif'*(f~i&s') — №'f'*QX'/s' m *e case °fa non-isolated singularity /
9 The monodromy theorem
9.1 Two parts of the monodromy theorem
(9.1.1) Let /: X —> 5 be the Milnor fibration of an isolated singularity,
and T: Hn{X,0, C) -> Hn(X,0, C), t0 ? S' be the monodromy transforma-
transformation. This is an andomorphism of a vector space of dimension ju. As is well
known, T can be reduced to a Jordan normal form, i.e. one can choose a
basis in which Г consists of Jordan blocks
/a i 0\
0
1
A/
where A is an eigenvalue of the monodromy on cohomology.
The monodromy theorem consists of two parts:
(i) the first concerns eigenvalues of T and asserts that all Я are roots of
unity;
9 The monodromy theorem
47
(ii) the second concerns the sizes of Jordan blocks and asserts that they do
not exceed n + I (we'll indicate a more exact estimate later).
In other words (i) means that the monodromy transformation Г is quasiuni-
potent, i.e. 3 integer N, such that TN is unipotent, and (ii) means that the
index of quasiunipotency of T, i.e. the least к such that (TN — E)k = 0,
does not exceed n + I, (TN - ?)"+1 = 0.
9.2 Eigenvalues of monodromy
Theorem I All eigenvalues of Гаге roots of unity.
Proof We give a beautiful proof due to Brieskorn [Br], based on the
regularity of the Gauss-Manin connection and on the positive solution of
the seventh Hilbert problem.
(9.2.1) Eigenvalues Ay of the monodromy Гаге algebraic numbers because
they are roots of a characteristic polynomial Д(А) with integer coefficients
(because Г originates from an automorphism of a lattice H"(X,0, Z)).
(9.2.2) The eigenvalues Ay are of the form Ay = e23""', where aj are
eigenvalues of the matrix Aq = — Reso V. Indeed, the characteristic poly-
polynomial A(/) of the monodromy Г = Г(/о) has integer coefficients, thus it
is constant under varying f0, and hence it is the same as for
To = e-^iv
(9.2.3) The positive solution of the seventh Hilbert problem @. A. Gelfond
and Th. Schneider, 1934) is given by the following theorem: If a and
A = e2ma are both algebraic numbers, then a € Q. Therefore it remains to
prove that ay are algebraic numbers. If they are, ay € Q and A = e2ma/ is a
root of unity. The algebraicity ofay follows from the algebraicity theorem.
(9.2.4) Algebraicity theorem of the Gauss-Manin connection. Let
x0, ..., х„ be coordinates on X at the point 0 and let / be a coordinate at
the point 0 € S. Then functions of (f s,o — C{ t} and of &x,o —
C{x0, ..., xn} are identified with converging power series. Now let
cp: С —> С be any automorphism of the field C. Applying cp to coefficients
of series, we obtain an extension of <p to an automorphism of functions
&x,o —* @xja- Let (pf be a function obtained from /in a such way. Denote

48
/ The Gauss—Manin connection
by Hf = H"(Q'X/SO) a germ of the De Rham cohomology, and similarly
for the function cpf. The algebraicity theorem asserts that the diagram
1(P v
Hvf-
where V is the Gauss-Manin connection, is commutative.
(9.2.5) It follows from this theorem that if a matrix Го = Res0 V is the
residue of the Gauss-Manin connection for f, then the corresponding
matrix (pTo for (pf is obtained by applying <p to all elements of Го. Hence,
if e2™"; is an eigenvalue of the monodromy for / then е2лг'*>(а-/) is the
eigenvalue of the monodromy for q>f, and consequently, е2л")>(а-') is also an
algebraic number for any automorphism ср. Now if ay were transcendental,
then we could transform a, to any transcendental number (p(aj) by some
automorphism cp: С —> С, and (because the set of algebraic numbers
is countable), we could choose cp such that e2mi>(.aj) js transcendental.
However, because e2m<f>(aj) is an algebraic number for any cp, it follows that
a.j is an algebraic number. This concludes the proof of the monodromy
theorem I. ¦
(9.2.6) Remark The theorem about eigenvalues of a monodromy was
conjectured by Milnor A968) based on calculation of A(A) for Pham
singularities. This theorem can be obtained from the corresponding theo-
theorem in the global case, when /: X —> S is a proper morphism. In the global
case the monodromy theorem was proved by A. Grothendieck, A. Landman
(Thesis, 1967), С. Н. Clemens A969) and others. As noted by P. Deligne
(see [Br]) the above proof (integrality of A(A), regularity, algebraicity,
seventh Hilbert problem) also gives a proof of the theorem about eigenva-
eigenvalues of a monodromy in the global algebraic case.
9.3 The size of Jordan blocks
(9.3.1) Theorem II (the second theorem about monodromy). If
fr+leJf = (f'Xo,...,f'Xn), then the size of the Jordan blocks of the
monodromy Tis at most r + 1.
(9.3.2) Remark Initially it was proved that the size of the Jordan blocks is
at most n + 1 (N. M. Katz, 1970). Brieskorn conjectured that the estimate
n + 1 is connected with the following result of J. Briancon and H. Skoda
9 The monodromy theorem
49
A974): For an isolated singularity, /n+1 6 J/. J. Scherk conjectured
theorem (9.3.1) in [Scl] and proved it in [Sc2]. His proof is based on an
embedding of the Milnor fibration /: X —> S in a projective family Y —> S
and on an application of the limit MHS on H"(Y,, C). We'll return to this
question in chapter 2, (II.3.5.9). Now we only note that the condition
f+l e J/ means that f'+1Qx^ С df AQX0 and is equivalent to the
condition tr+l-.J^f) С -^о^-
When r = 0 the condition / 6 J/ is equivalent to quasihomogeneity of
a singularity f, and the monodromy theorem asserts that all Jordan blocks
are of size 1, i.e. the monodromy T is semisimple. We have already
obtained this in E.5.5). B. Malgrange [M2] gave an example of singula-
singularities who monodromy T contains a Jordan block of maximal possible size
n + 1. Such an example gives the singularity л:8 + y% + z% + x2y2z2 for
which N2 фи (see also [S3] and (II.8.5.15)).
7
9.4 Consequences of the monodromy theorem. Decomposition of
integrals into series
Let Л = {Alt ..., Xr} be a set of eigenvalues of monodromy T on the
cohomological fibration H= R"f'*CX' = U tes'Hn(X,, C). Then A^\
..., A~' are eigenvalues of the monodromy Mon the homological fibration
if = U tes- Нп{Х„ С) which is dual to Я Denote by
(9-4.1) «У = -гк^
the eigenvalues of a matrix R, M = е2л'я. By monodromy theorem I (§9.2)
the eigenvalues A, = e~2jria> are roots of unity, and so ay- 6 Q are rational
number defined mod Z. Denote by
(9.4.2) L(Aj) = {a.j, a.j + I, a^ + 2, ...)
an arithmetical progression with one (suitable) value of a;.
Let ?Oi, ...,а)м be a basis of (local) sections of the sheaf
3% = Я® 6>s\ and let y(t) be either the local or multivalued section of
the fibration ff, i.e. a horizontal section of the sheaf .Ж* =
Я* ®сс. @s'- Then the column
> = Qy@= I ал, ..., шЛ
\iY(t) Jy(O J
is a solution of the system y' = T'(t)y (8.1.5), where Г@ is the matrix of
coefficients of the Gauss-Manin connection V on J& in the basis
coi, ..., co^. By G.3.4) we have у = Z(t)tRC, where e2MR = M and С is

50
/ The Gauss—Martin connection
a column of constants. By the regularity of the Gauss-Manin connection,
the matrix Z(t) is a matrix of functions meromorphic at the point 0 € S.
From the structure of the matrix tR G.3.6), we obtain:
(9.4.3) Corollary If со is a section of the sheaf. ffl and y(t) is a horizontal
section of the homological fibration //*, then the integral
7@ = IY,o, = f со
Jy(t)
is decomposed into the series
(9.4.4) 7@ = У]
1
k=o K-
and, moreover, if for an eigenvalue Я the monodromy Г (or M) has Jordan
blocks of dimension =s r + 1, then the inner summation goes only up to r,
(9.4.5) Corollary If in (9.4.3) со is a section of the sheaf M(~x) in a
neighborhood of the point 0 € S, or ш is a holpmorphic и-form in a
neighborhood of the point 0 € X representing this section, then in the
decomposition (9.4.4) all a s* 0 (all progressions L(kj) consist of non-
negative numbers), and, moreover, if a — 0, then a^a = 0 for к s= 1.
Proof This follows from Malgrange's theorem F.2.1), where it is proven
that lim,^o 7@ = 0. ¦
(9.4.6) Corollary If in (9.4.3) со is a section of the sheaf J&m in a
neighborhood of the point 0 € S, or со is a holpmorphic (n + l)-form in a
neighborhood of the point 0 € X representing this section, and w = s[oj] is
a geometrical section of the sheaf .^ corresponding to the Gelfand-Leray
form co/d/ = Res [co/(f — t)], then in the decomposition (9.4.4) for
integrals
7@
-j „-j
Jy(O Jy
0,/d/
Proof By proposition E.3.4) the operator 9,: ^"'^^fC) is an iso-
isomorphism. Let [со] = dt([r]]), where rj is an л-form. We have calculated
the derivative of the function F(t) =
10 A non-isolated hypersurface singularity
51
In the decomposition of F(t) all a 3= 0, and for a = 0 we have only
constants by (9.4.5), hence in the decomposition of I(t) = F'(t) all
10 Gauss-Manin connection of a non-isolated hypersurface
singularity
Let /: (C+1, 0) —> (C, 0) be any, not necessarily isolated, singularity of a
holomorphic function / In this section we'll consider how the notion of
the Gauss-Manin connection of the cohomology of the Milnor fibration,
the sheaves ^f(-/) and other notions and results obtained above for isolated
singularities are transferred to a general case [Sr].
10.1 De Rham cohomology sheaves
A0.1.1) The Milnor fibration /: X —* S of a singularity / is defined word
for word as in the case of an isolated singularity [Di3]. Xis an intersection
of a ball В = {\x\ <e} С Cn+1 of radius ? with the preimage of a disk
5 = {|f|<E}cC of radius <5 < ?. Here Xo = f~l@) is the single
singular fibre. The morphism / is a smooth locally trivial fibration over
S' = S\{0}, /': X' -> S'. It defines cohomology fibrations № -* S',
where Hp> = \J t&s'Hp(Xt, C) = Rpf*CX', denning topological Gauss-
Manin connections V on locally free sheaves of sections
Ж" = Hp <g> &S' = RpftCx- О ^V-
The monodromy transformation Г corresponding to a standard generator of
Я!E') acts on a fibration Hp. The Betti numbers bp = dim Hp(Xt, C) are
called Milnor numbers /г р of the singularity /.
A0.1.2) By the relative Poincare lemma the sheaf f~x@s' has a resolution
Q'X'/s'> a complex of sheaves of relative differential forms. Thus sheaves
Л?р are identified with sheaves of relative De Pham cohomology
Су
These sheaves are naturally extended to sheaves J%f?R(X/S) =

52
/ The Gauss-Manin connection
Rp'/'*(QX/s) on S, which we'll denote by -Ж^_2) in analogy with the case
of isolated singularities.
A0.1.3) The morphism/(as well as/') being Stein, these sheaves
are isomorphic to the cohomology sheaves of complex f*Q'x/s- Thus we
obtain (De Rham theory) a connection between the sheaves ^fp and
sheaves of differential forms (the connection of topology with algebra and
analysis).
10.2 Coherence
A0.2.1) There is a question about the coherence of sheaves
This is one of the fundamental technical results. Since it is technical, we'll
not consider this question here and we'll assume that the singularities
under consideration are such that these sheaves are coherent. We mention
only the papers of Hamm [H], R. O. Buchweitz and G. M. Greuel A980),
and also that of van Straten [Sr], in which Brieskorn's proof of the
coherence of sheaves .^f^K(X/S) for isolated singularities is transferred to
the case of non-isolated concentrated singularities. We restrict ourselves by
a definition of this notion.
A0.2.2) Let X -?¦ S be the Milnor fibration, where X = XE<&, with s and д
radii of a ball and a disk, respectively. Denote by dX = dB? r\f~l(Sj) and
X = X U dX the relative boundary and closure of X. A sheaf L of C-
vector spaces on A"is called transversally constant (relative to the boundary
dX), if it is constant along transverses to dX. This means that there exists
an open neighborhood U of the closure dX in CN and a C00-vector field 9
on C/such that: A) в is transverse to dB; B) the local 0-flow in [/leaves X
and the fibres X, invariant; C) the restriction of L to the integral curves of
в is a constant sheaf.
Consider a finite complex (K\ d) of ^V-coherent modules with
/~1(^sHinear differential d. Van Straten proved that if the cohomology
sheaves .Ж'(К-) are transversally constant, then the sheaves U'f*(K-) are
бVcoherent modules.
A complex of sheaves (K\ d) is called concentrated if for all e' ? @, e]
there exists d' € @, d] such that the cohomology sheaves .Ж'(К') are
transversally constant on AV,<y- A singularity /is called concentrated if the
complex Q'x/s 1S concentrated. Examples of concentrated singularities are:
10 A non-isolated hypersurface singularity
53
A) isolated singularities; B) quasihomogeneous singularities (with a good
С -action); C) singularities such that for some X —> S there are only
a finite number of isomorphism classes of germs among germs
(X, x) -»• (S, f(x)). Note that the sheaves .%f'DK(X/S) may be non-coher-
non-coherent, as is the case, for example, for/ = yA + xy2z2 + z4.
10.3 Relation between .%/p(f*Q-f) and j\Л''f'(Qy)
A0.3.1) Proposition C.6.1) can be generalized to the case of non-isolated
singularities. As in the case of isolated singularities, from the second
hypercohomology spectral sequence C.2.4) we obtain an exact sequence
containing the De Rham cohomology sheaf .%?'DR(X/S) = .Jtf'if+Qy),
A0.3.2) 0 -* (Л'ДС*) ® ts -* jy'CAQy) - U&i&f) - 0
Cx J J
where the first sheaf has a zero germ at 0 e S, and the last one is
concentrated at 0 € S because J^'(Qy)\S- = 0 by virtue of the smoothness
of/over S'. It follows that Jr'(QyH ^ f*.%?'(Q-f)o.
If/ has a concentrated singularity at the point x0 = 0 e X, then
A0.3.3) /*(.Ж'(?2у)H ~ ЖЩ)* * НЩЛ)
Van Straten [Sr] adapted Brieskorn's proof of this fact for isolated
singularities to the case of concentrated singularities.
A0.3.4) If the De Rham cohomology sheaf Ж'(/*Qy) is torsion free,
then it is locally free of rank fij and we get a formula for the calculation of
Milnor numbers
Below we'll obtain a criterion for the sheaves .M'(f*Qy) to be torsion
free.
10.4 A general method of extension of a singular connection over the
whole disk
A0.4.1) The topological connection on the sheaves :MP = Rpf*Cx'
<8>Cjr @s is carried to the sheaves .^?qR under the isomorphism
:%" ~.%$>K(X'/S'). The calculation of this connection V on
¦^mW/S') is based on the Leray residue theorem and this was carried
out in §4. We obtained the following result. If a local section [a>] of the
sheaf .Ж^Х'/S') = .%fp(f*Q-r/s-) is represented by a /?-form со of A"
then d,([w]), d, = Vd/dr: JffpR(X'/S') ->¦ JSfgR(*'/?'), is defined in such
a way. We take the form dci>; w represents a cycle in Qx,,s,, and therefore

54
/ The Gauss-Manin connection
dco = d/ Л т], where tj e Qx. Then df /\dr] = 0 and hence d?7 = d/ Л ?
by the division lemma (morphism/' is smooth), i.e. tj is a cycle in ?2>y5'.
The class [77] e ЛС^К{Х'/S') is correctly defined and just this class is
<9,([a>]) = [77]. Note that [rj\ = dco/dt € JV^X'/S') is a derivative of <w
in t in the sense of Leray.
A0.4.2) We can interpret the operator d, as a connecting homomorphism
in an exact cohomology sequence. Namely, there is an exact sequence of
complexes
0-+Q- d2?Q-
by virtue of the smoothness of/': X'
A0.4.3) 0-+/*Q>-' "ЛЛ.
0
S'. The sequence
О —i f O-*3*
¦"Л" "-* J *^?X'/S'
¦0
is exact since /' is Stein. The sheaf .Ж^К(Х'/S') is the cohomology sheaf
.3%p(f*Q,X',S'). A choice of a p-form со € /^Qj, representing the class
[со] e Jg^X'/S') is a lift of this class from f*Qx,/s,) to /*Q?,. We see
that taking dco, etc. is exactly the description of the construction of the
connecting homomorphism <5: J
A0.4.4) Now it is clear how to extend the Gauss-Manin connection over
the whole disk. We have to extend the exact sequence of complexes
A0.4.3) to an exact sequence over S and then the connecting homomorph-
homomorphism in the exact cohomology sequence is an extension of the Gauss-Manin
connection dt.
A0.4.5) A meromorphic Gauss-Manin connection V on the sheaf
¦%p[0] = J^?p(/"*Qy)®r.s <^s[0] is obtained if we take a connecting
homomorphism dt coming from the exact sequence
The regularity of the connection singularity at the point 0e5 can be
proved as in (8.3.1)-(8.3.6).
We proceed (see [Sr]) to the construction of sheaves .^^./) on 5 - the
analogs of the sheaves .^'"'' in the case of an isolated singularity /
10.5 The sheaves -%>(-\) and the Gauss—Manin connection dt:
A complex Q> = QV/<r of sheaves of relative differential forms is defined
' X'S d/Л
at the Coker of the homomorphism Q' —> Q', where (Q" = Q^, d) is the
10 A non-isolated hypersurface singularity
55
De Rham complex of differential forms on X. Let (Q', d/) be the Koszul
complex of a function /
A0.5.1)
n
0
Q
n+\
0
This is really the Koszul complex (K\f) constructed from the sheaf
Q^~<$J+1 and the sequence of partial derivatives /,=/.v,,
i=0, ..., n. Denote by 5' = Im (Q- d-^A Q) = d/ Л Q' the complex of
sheaves of Koszul coboundaries. We have an exact sequence of complexes
0 -» B- -» Q- -» Qy -» 0
and we get an exact sequence on S
A0.5.2) 0-»/*?• ->/*Q- ->/*Q)-->0
because/is Stein. This sequence extends the exact sequence A0.4.3) over S.
We set
ПО 5 3^ Wp Wp(f O* ^ = i$^p (X~I*}Л
In the case of an isolated singularity the extension of the Gauss-Manin
connection dt: .Ж(~2) -> :Ж(~Х) = f*Q"f/d(f*Q"f]) is found to be a
connecting homomorphism in the exact cohomology sequence for A0.5.2)
and by identification (using the division lemma) of 5' with Q^ ,s and the
inclusion В' С Q' with Q^ ,s —* Qx. Therefore by definition we set
A0.5.4)
and the Gauss-Manin connection d,: -^l-%) ~* ^%(-\)t0 ^e a connecting
homomorphism д in the exact cohomology sequence corresponding to the
sequence A0.5.2). The next local description of d, shows that dt really is a
connection, i.e. the Leibnitz identity is fulfilled.
A0.5.5) If [со] e .%?p(f*Qy) = Эв[_1у where со € Qp, then
3,(M) = [do,] € .^"+1 (/*Я0 = Щ_ху
where dco e f*B- and can be presented in the form dco = d/ Л tj, tj e Qp.
A0.5.6) Lemma dt:
^_X) is an isomorphism.
Proof Using the exact cohomology sequence corresponding to A0.5.2), we
obtain

56
/ The Gauss-Manin connection
Since X\s smooth, the complex Q" is a resolution of the sheaf C^, i.e. it is
acyclic for p э= 1. Since / is Stein, the complex /*Q' is also acyclic for
10.6 The sheaves 3%?Q) and the Gauss-Manin connection d,:
@)
For an isolated singularity the sheaf .Ж/@> = Q"+1/d/A d(Q"-') coin-
coincides with д,(.Ж/(~])). Therefore in order to generalize sheaves Ж(й) to the
case of non-isolated singularities following our recipe in A0.4.4), we have
to extend the exact sequence A0.4.3) over S to have the complex
B- = d/ Л Q~' on the right. Consider the exact sequence of complexes
A0.6.1)
where
A0.6.2)
Z-
Z" =
0,
is the complex of sheaves of Koszul cocycles. As the morphism /is Stein,
we have an exact sequence of complexes on S
A0.6.3)
0.
Consider the exact cohomology sequence corresponding to A0.6.3)
A0.6.4)
Therefore, by definition we set
(Ю.6.5)
)
and the Gauss-Manin connection d,: Ж?_Х) -* Ж^0) to be the connecting
homomorphism in the exact sequence A0.6.4). We can also see that d, is
really a connection, i.e. the Leibnitz rale is fulfilled, from the following
local description of d,.
A0.6.6) If [w] € .MP+x{f*B-) = .i?f(% and со = dfArj (rj is a 'lifting'
ofo> to /*Q- in A0.6.3)), then
d,№) = d,(W Л r,}) = [Arj\ G Jg" +i(f* Z-) = .^f0).
Since .%fP(f*Q-) = 0 for p > 1, we obtain from A0.6.4):
A0.6.7) Lemma д,: -Ж^^.Ж^ is an isomorphism.
10 A non-isolated hypersurface singularity 57
10.7 A generalization of diagram E.3.4)
We can join the exact sequences of complexes A0.5.2) and A0.6.3) in one
big diagram.
A0.7.1) Definition The sheaves Ж? = .3%p(Q\ d/Л) are called the
sheaves of Koszul cohomology of the singularity / We have a complex
(Ж', d) of sheaves of Koszul cohomology
Ж' = Z-/B- =Ker(Q-^Q-)/Im(Q-^Q-).
Obviously we have a commutative diagram of complexes with exact
lines and columns
1
в-
?•
0
T
Td/Л
СЙ' —>
и
cz- -»
T
0
0
T
Td/Л
Qy -»
и
Ж' —
т
0
0
• 0
A0.7.2)
The morphism /: X -* S being Stein, we can apply the functor /* to
A0.7.2) and obtain the diagram of/* A0.7.2) (in which all the sheaves are
exchanged with their direct images), in which the lines and the columns
remain exact. The exact sequences A0.5.2) and A0.6.3) are the exact
sequences in diagram /* A0.7.2) which go through the 'centre' /*Q\
Now we consider two exact sequences which 'go' through /*Ж\ These
give us the connection between the sheaves Ж^_^, i = 0, 1,2.
Write the exact cohomology sequence corresponding to the sequence
0 —»/+ В' —> /* Z' —> /*Ж' —> 0 in the upper line and the one corre-
corresponding to 0 -* /*.Ж' -»¦ f*Q
following commutative diagram
A0.7.3)
0 in
f*B
'+l
0 in the lower line of the
я-'
it
Л'
¦* '(/,«>)•

58 / The Gauss--Manin connection
In this diagram the homomorphism
A0.7.4) MV(_X) = .%f+1(f*B-)±
is induced by the inclusion f^B' —> f*Z\
A0.7.5) [2) y
+1
y >
is induced by the epimorphism f*Qy —> f*B+1, and д denotes the
connecting homomorphisms. Moreover, in diagram A0.7.3) we included
the operators
A0.7.6) аг1: .Ж(% - i??(%, i = 0, 1, 2,
which make it commutative, and C-linear homomorphisms
A0.7.7) a:
A0.7.8) Proposition The following conditions are equivalent:
(i) the homomorphism a = 0;
(ii) one of the homomorphisms d~x (and thus all of them) is a mono-
morphism;
(iii) i or j are monomorphisms.
Proof This is obvious by virtue of the exactness of lines in diagram
A0.7.3) and the fact that dt are isomorphisms.
A0.7.9) Corollary If a = 0, then diagram A0.7.3) falls to the following
pieces with exact lines
A0.7.10)
Ад. \?'~' Ад,
Diagram A0.7.10) generalizes diagram E.3.4) to the case of non-isolated
singularities.
A0.7.11) In [Sr] Van Straten proved that -i?^,) are ^'s-coherent modules
if/is a concentrated singularity. He also obtained the following criterion.
Theorem The condition a — 0 is equivalent to the absence of torsion in the
sheaves -J^^,y
10 A non-isolated hypersurface singularity 59
The proof of this consists of carrying Malgrange's proof of §6 to the case
of non-isolated singularities.
Van Straten [Sr] verified the condition a = 0 for the non-isolated
singularities with a one-dimensional critical set, which is a reduced
complete intersection. For these singularities
/л"'1 = 1 or 0
and
ixn = dim Нп+\.Ж-) = dim(Q"+1/d/ Л Q" + &Ж").
This formula generalizes the celebrated formula for the Milnor number of
an isolated singularity [Mi]
jx = dimQn+1/d/ л Q" = &m@c»,fi/Jf.

II
Limit mixed Hodge structure on the vanishing
cohomology of an isolated hypersurface
singularity
1 Mixed Hodge structures. Definitions. Deligne's theorem
We first recall very briefly the necessary basic definitions (for a more
detailed survey see, for example, [K-Ku].
1.1 Pure Hodge structure
A.1.1) Definition A pure Hodge structure, or simply a Hodge structure
(abbreviation HS) of weight n is a pair {Hz, Hp-4} consisting of a lattice
Hj of finite rank and a direct sum decomposition of the space H =
Я =
p+q=n
such that
HM ="~
The motivation for introducing such a concept lies in the Hodge
decomposition into (p, ^)-types of the cohomology of a smooth projective
variety or a compact Kohler manifold.
A.1.2) Defining the HS in the form of a Hodge decomposition
{Hi, Hp'4} is equivalent to giving the Hodge filtration {Hz, F-], where
F' is a finite decreasing filtration of H such that
H = FP@ F"~P+l for all p € Z.
The filtration F' defines a Hodge decomposition by
A.1.3) H"'4 = fnR
The Hodge decomposition defines the Hodge filtration by
60
A.1.4)
F" =
1 MHSs
Hr,n-r = Hp,n-p Q
61
i.e. F" = Hn'°, F" = Я"-0 ® Я"-1, ...,F° = H.
In what follows we'll write an HS by its filtration F* (assuming that a
lattice Hz is fixed).
1.2 Polarised HSs
A.2.1) Definition A polarized HS of weight n is is an HS {Hz, F'}
together with a non-degenerate integral bilinear form
Q: HZX Нж-> Z,
symmetric, if n is even, and antisymmetric, if n is odd, and satisfying the
Riemann bilinear relations:
in terms of a decomposition
A.2.2) Q(Hp'", Я>'-*') = 0 for (//, q>) ф (р, q),
ip~4Qbp, V) > 0 for VV e Hp-\ грфО
or in terms of a filtration
Q(Fp,F"-p+l) =
A.2.3)
where С: Я -* Я is the Weil operator C| WM = i"~q.
The bilinear form g is called a polarization of the HS F\ The origin of
this notion is the bilinear form Q(q>, ip) = (—1)"("~1)/2 \xLd~" Л ср Л i/» on
the space of primitive n-cohomology of a smooth projective variety of
dimension d.
1.3 Mixed Hodge structure
A.3.1) Definition A mixed Hodge structure (abbreviation MHS) is a triple
(Hz, W,, F'), where Я/ is a lattice, W. is a finite increasing filtration on
Hq = Hz <8> Q, and F' is a finite decreasing filtration on Я = Яг <g> С,
such that the induced filtration on the quotient Gr^H = Wk/Wk_\ defines
an HS of weight k. Here F^Gr^H is the image of Fp П ^Я in Grf H,
FpGrwkH = ^Я r\Fp+ Wk_xH/Wk-X H.
F~ is called the Hodge filtration and W. the weight filtration. The notation
for an MHS may be abbreviated to: a pair of filtrations (W,, F).
A.3.2) Let Нр'Ч = GrpFGrJ+qH. Then Grf = 0,+,=* Я^«. The num-
numbers hp-4 = dim Яр-« are called the Hodge numbers of the MHS (W., F-).

62
IILimit MHSs
A.3.3) If there are two MHSs on spaces Яand H', then the space of linear
mappings Horn (Я, Я') is endowed with a natural MHS,
fF;nHom(tf, #')q = {<p: HQ -> H'u\cp{WkH) С Wk+mH',Vk},
FpKom(,H, H') = {q>: H-+ H'\cp(FkH) С Fk+PH', Vk).
It is easy to verify that Нот(Я, H')™ = © Нот (#'•*, H'r+p-s+i). In
particular, a morphism cp: H —> H' is called an MHS morphism of type
(r, r), if ф(Я2) С #i, <р(Ж*Я) С ^+2гЯ' and cp(FpH) С F^+^'; <р
is called an MHS morphism if r = 0.
One of the most important properties of an MHS morphism is its strict
compatibility with both the filtrations F- and W_, i.e.
A.3.4) cp(WkH) = Wk+2rH' Dlmcp, cp(FpH) = Fp+rH'Dlm<p.
A.3.5) It implies that if H' ->• Я ->• Я" is an exact sequence of MHSs,
then it remains exact after application of Grf, GrpF and GrpFGr^.
1.4 Deligne's theorem
A.4.1) Theorem The complex cohomology groups H'(X) of projective
varieties/C carry MHSs which are functorial. In the case of a non-singular
variety this MHS reduces to the ordinary HS of pure weight.
When proving this theorem Deligne introduced the filtrations W. and F"
on the level of resolutions 0 —> C* —» K- of a constant sheaf C* (or Cq),
which induce filtrations on the hypercohomology Hi(^r, K~) ~ Hk(X, C).
2 The limit MHS according to Schmid
2.1 Variation ofHS: geometric case
Let ж: Y —> S be a family of smooth projective varieties Y, =
я"'@ С Pm, parametrized by a complex manifold S (У С S X Pm and л
is induced by a projection). Then cohomology groups of fibres H"(Yt, C),
0 « n «s 2d (d = dim Yt), are glued to a locally constant fibration (local
system) H=R"n*CY ={J tesH"(,Yt, C) on S. Let .%? = &S(H) = H
®cs ^s be a sheaf of holomorphic sections of this fibration. The local
system H<ZM defines a connection V on .^f. This connection is called
the Gauss-Manin connection of the family л. The local system Я contains
a real subsystem Нц = К"л*Шу and even a sublattice Hz, fibres of which
are images of Hn(Yt, Z) in H\Yt, U). There is a Hodge filtration (a flag)
F; on each fibre H, = Hn(Yt, C). Griffiths showed that this family of flags
2 The limit MHS according to Schmid 63
depends on t holomorphically. We obtain a decreasing filtration of holo-
holomorphic subsheaves in a locally free sheaf Ж
... С .Гр+Х С .Гр С ... С Ж.
The Griffiths transversality theorem asserts that V(.5^p) С .9'р~х ® Q^.
This geometric situation motivates the following definition.
2.2 Variation ofHS: definition
B.2.1) Definition A variation of Hodge structure (VHS) of weight n on a
complex manifold is a pair (Я, .T'\ where Я is a locally constant fibration
(or a locally free sheaf Ж with an integrable connection V, Ker V = Я)
defined over U (or Q), Я = Нц <g> С, and a decreasing filtration
.^- = {.^p} of locally free subsheaves i^ С Ж = H<g>Cs @s- These
objects have to satisfy the following two conditions:
(i) for each point t € S the filtration .9~' induces a filtration .9'\ on the
fibre H, at t constituting an HS of weight n;
(ii) V(J^) С Q^ ®^x .^^-' for each p.
B.2.2) Moreover, if there is a flat, non-degenerate bilinear form
Q:HXH-*Qs defining a polarized HS, then the VHS (Я, .T') said to
be polarized.
If in the geometric case we consider P = \J /е5Р"(Уь С) instead of
И^= U tesH"(Y,, C), where Р"(У/, С) is a primitive cohomology, then we
obtain a polarized VHS. It is convenient to consider the polarized VHS for
technical reasons. However, an HS on Hn(Yt, C) is completely defined by
HSs on Р*(Уг, С) and so no information is lost.
2.3 Classifying spaces and period mappings
B.3.1) Let JT be a set of all filtrations (flags) F- in a vector space Я of
fixed dimensions fp = dim Fp = J2r^Phns. It is a closed subvariety of a
product of Grassman manifolds. Let Q be a bilinear form on Я defining a
polarization. Then the set of polarized HSs of given type is parametrized
by the set of F- ejf satisfying condition A.2.3). The first bilinear
relation defines a closed subvariety D С JF, and second bilinear relation
defines an open subset D С D; D is called a classifying space of the
polarized HS of given tjpe.
B.3.2) Let G={g6
, go) =
), Vii, о € Я} be an

64
IILimit MHSs
orthogonal group of the bilinear form Q, and Gr С GL(Hu) the corre-
corresponding real subgroup, Gj_ — {g € G^\gHi = Hj\ an arithmetic sub-
subgroup of Gr. The group G transitively acts on D, D ~ G/B, and the group
Gr transitively acts on D, D = Gr/ V, V = Gr П В. For clarity it is useful
to have in mind the simplest example of an HS of weight n = 1,
Я = Я0-1 ф Я1-0, A0-1 = Au0 = 1. In this case Gc = SLB, C), Gr =
51B, R), D = Р'(С), and D = U С Р'(С) is the upper half-plane.
B.3.3) The group Г = Gi acts on D properly discontinuously. Hence
there is a structure of a normal analytic space on D/Г; D/T is called a
period space of a polarized HS of the given type.
B.3.4) Now assume that there is a variation of the polarized HS (Я, .У')
on a manifold S, for example a VHS corresponding to primitive n-
dimensional cohomology of a family of projective varieties л: Y —> S. Let
e: S —> S be a universal covering. We can lift the VHS to S: e~xH =
HX s S = His a locally constant fibration on 5" defining a connection V
on the sheaf.Ж = e*.M and &"' = e*.?~' is a decreasing filtration on 3@.
Since S is simply connected, V defines a canonical trivialization of the flat
locally constant fibration H, Я ~ H X S, where Я is a vector space (a
canonical fibre). This trivialization defines a family of filtrations
F-s с Hs ~ H on the space H{Hs~Ht,t= e(s), F's = F;). We obtain a
mapping Ф: 5 —» D, 5 н-> F's to a classifying space D. If 5 and 5' € S,
e(s) = e(s') = t, then it is easy to see that F-s and F-s, differ by an element
у € Г, F's< = yF-s. Hence, the mapping Ф is reduced to a mapping Ф:
S —> D/T; Ф is called a period mapping of a given variation of Hodge
structures.
We'll be mainly interested in one-dimensional variations of the HS. Let
us consider more carefully a period mapping in this case.
2.4 The canonical Milnorfibre
B.4.1) Let (Я, ,9~-) be a variation of HS over a punctured disk S' = S\0,
5={|^|<1}, where .T' is a decreasing filtration on the the sheaf
Ж = Я<8>с5. (9s1- Let V be a connection on :%f defined by the locally
constant fibration H = Ker V.
In a geometric situation we have a projective morphism л: Y —> S,
which is smooth over S', and H = В."п*Сг = (J ,€х'Ял(У;, С) (to have a
polarized VHS we need to consider primitive cohomology Pn{Yt, C)
С H"(Y,, C) instead of H"(Y,, C)). The connection V on the sheaf
2 The limit MHS according to Schmid 65
s' = U^^Qy4s,) = .^R(Y'/S') is the Gauss-Manin
connection.
B.4.2) Denote by ht: Y, —> Y, the Picard-Lefschitz monodromy cone-
sponding to a canonical generator of Л\ (S't t). This is a diffeomorphism of
a fibre Y, = n-\f), t ф 0. Let
and
Г = (А*)-1: H"(Yt) - Hn(Yt)
be, respectively, monodromy transformations for homology and cohomo-
cohomology, T = (M* Г'. ы
B.4.3) To define a period mapping we have to identify fibres at different
points t € S'. This can be done by fixing a fibre Ylo, t0 € S', and
transferring filtrations to this fibre. To avoid dependence on the choice of a
point /0 and to make the construction more invariant, we introduce the
notion of a canonical fibre Y^.
Let U = {u e C|Im и > 0} be the upper half-plane
B.4.4) e:U-*S',u
= e(u) = e2jri"
df
be the universal covering of S'. Denote by
Г = л'\Б')тге
an inverse image of
B.4.5)
i
U э и
X
s
Ye(u)
SDS'^UBu ,Ш = (х,и).
Denote by j'u: Y, с Yx an embedding of the fibre Y, onto a fibre over a
point и € U, ju(x) = (jc, и) б Уоо, ^ = е(м). Here У^ —» U is a locally
trivial fibration with fibres diffeomorphic to Y,. Since U is contractible, the
embeddings /„ are homotopy equivalences and, consequently,
B.4.6)
Hn(Yt)
04*
), Я =
are isomorphisms. Via these isomorphisms we'll identify Я"(У,) with a
fixed space Я = Я"(Уоо)- For this reason Уо, is called a canonical fibre of
a family jr.
For an abstract variation of the HS we have
B.4.7) К*-Ки = е-1Н=НХ.
S't-U
h

66 II Limit MHSs
B.4.8) We obtain an identification Я = H°(U, Ни)^Н, of the space Я
with a fibre H,. We can consider the space Я as a space of (single-valued)
horizontal sections of the fibration Ни or as a space of multivalued
horizontal sections of the fibration Я The identification j*: H-=iHt
consists of a restriction of a section A e H = H°(U, Hjj) to a fibre
Hu = Я/, t = e(«). The inverse isomorphism consists of an extension of
At ? H, to a multivalued section Л over U with the initial condition
A(u) = A,; #also is called a canonical fibre of the fibration//.
We obtain a canonical trivialization
B.4.9)
5"
х u,
, и).
B.4.10) Let us carry the monodromy T: H, —»H,, which exists on every
fibre Hj and is obtained by parallel transference along the canonical
generator of n\(S', t), to the canonical fibre Я by means of the identifica-
identification
Consider a diffeomorphism
B.4.11) h: Ни -* Ни, (со, и) н-» (о>, и + 1),
and the corresponding linear transformation
transferring a multivalued section A such that A{u + 1) = Ao to a multi-
multivalued section h*A such that h*A(u) = Ao. Obviously, this implies:
Л
B.4.12) Lemma Under the identification H-^H, for any и the mono-
monodromy T on H, corresponds to the transformation h*, i.e. there is a
commutative diagram
h*A?H&H,
A € H ^ Н, Э Ао.
f.
Proof. Let Ao G Я,. Then (/*)~'(^o) = Л € Я is a multivalued section
such that Л(и) = Ao or A{u + 1) = 7M0- By the definition of A* we have
h*A(u) = A(u + 1) = 7^o, and this means that h* A = (/*)""' (TA0). ¦
2 The limit MHS according to Schmid 67
Thus, on the canonical fibre Я the monodromy T = h*. Moreover, two
identifications j* and j*+x of the canonical fibre Я with Я, are connected
with the monodromy on Я in the following way: there is a commutative
diagram
B.4.13) Я Jt
h* = T
H *t
The proof is obvious: if Ao € H,, then (/*+i)~'^o = A is a multivalued
section such that Л(м + 1) = Ao, and this means that h*A(u) = ^0, i-e.
2.5 The Schmid limit Hodge filtration F's
A variation of the HS (Я, .^') defines a family of filtrations F\ on fibres
Ht, t G 5'. We can lift this family onto U and by means of a trivialization
B.4.9) Hjj ~ Я X t/ we get a family of filtrations F-u on the space Я
B.5.1) u^F-u = U*ur1F;,t=e(u).
We obtain a period mapping
B.5.2) U-?> D, и >-> F-u.
B.5.3) Lemma The mapping ф satisfies a condition
Proof. We have Д, =.? A* B.4.13), hence
, = T-'F
If we take a quotient of D by a cyclic group {Г*}, we obtain a period
mapping tp
B.5.4) U-^D
4 -I
S'-^D/T
1
By the monodromy theorem ([Sm] and Theorem I of §1.9.2 all eigenvalues
of Гаге roots of unity. Hence, if T = TSTU is a Jordan decomposition,
where Ts and Tu are semisimple and unipotent parts of the monodromy,
then there exists m such that 7\m = 1.

68 II Limit MHSs
Let T — е2л[к. By means of a family of automorphisms g(u) = е2отЛ" of
the space Я having monodromy T, g(u + 1) = Tg{u), we can 'untwist' the
mapping ф and make it periodic: set гр(и) — g{ii)y{u), then гр(и + 1)
= g(u+ \)-ф(и+ 1)= g(it)TT~l ф(и) = гр(и) and hence, гр is 'des-
'descended' to 5",
B.5.5) U -X D
S' ^ , г~р{и) = е2-т'л"-ф(м).
A matrix R{ = (l/2m)log Г) such that T = e2jtiR is not defined un-
uniquely, more exactly its semisimple part is not defined uniquely. Let
R = —Rs - N, where -Rs is a semisimple part, and —N is a nilpotent part
of R. Then Г = eb*(-R.-N) = ^.^ j e ^ = e-2jri^, Ги = е~2"ш. The
nilpotent operator
B.5.6) N= rlogru,
the logarithm of the unipotent part of the monodromy (divided by —2m), is
defined by T uniquely. Here for an unipotent operator Tu we set by
definition
B.5.7)
log Tu = log [E + (Ta -E)] = X>
B.5.8) Theorem-definition Let (Я, .Я") be a variation of an HS over a
punctured disk 5", let T = TsTa be the monodromy and let ?m = 1.
Consider a covering 5" —> S', t = ~tm, killing the semisimple part of the
monodromy. Consider a mapping гр: S' —> D, defined for a variation of HS
on S' in B.5.2) (to simplify notation we can assume that the monodromy
T = Tu is unipotent, R = —N). then the mapping гр: S' —> Z) can be
continued holomorphically over the point 0 ? S. The point гр(О) g Z) is a
fixed point of Ts. The filtration F$ = F-x corresponding to this point is
called a limit Hodge filtration (according to Schmid). By construction F's
is invariant wrt Ts.
A generalization of this theorem for the case dim 5 > 1 is part of
Schmid's nilpotent orbit theorem [Sm]. In our one-dimensional case this
theorem follows from the regularity of the connection V A.8.2.1). Indeed,
the regularity of a connection means in essence that гр is meromorphic, i.e.
it has at most poles (see below). Then the map гр is regular at the point
0 € S because D is a projective variety (we can 'get rid' of denominators).
2 The limit MHS according to Schmid 69
2.6 An interpretation ofFs in terms of a canonical extension of Ж
B.6.1) It is known that any locally free ^л-module Ж on a punctured
disk iS" is free. Trivializations of the sheaf Ж are in one to one
correspondence with extensions .Ж of the sheaf.X over the whole disk S:
if a trivialization of .Ж is chosen, i.e. a basis of sections a>\, ... , соц of
Л? over S', then the fibre Жо consists of those со = YlsA^j, for which
gj(t)€^s,o- A sheaf Ж = Я ®cs, б s' with a connection V has a
canonical (privileged) extension over S. We'll denote this by Z. We'll
consider the canonical extension in more detail in §6. Now we assume that
the monodromy T = Ги is unipotent, and let T = e-2;lW, N =
-a/2*ri)logrn.
Let A g Я, i.e. A — A{t) is a multivalued horizontal section of Я (or
A = Л(е2лгш) is a horizontal section of Hv). The generator у е n\(S, t)
acts on A by means of monodromy yA(t) = ^(e2jti/) = Щ/) Consider a
family of automorphisms 'untwisting' multivalued sections
g(t) =
= /" loe'
or
= e2jri№,
B.6.2)
where w = (l/2tti)log /. Then у transforms g{i) to e2*1"g(t) = Г
Therefore, sections
B.6.3) s = s[A] = ^@
are invariant wrt у and, consequently, define single-valued sections of the
sheaf Ж.
B.6.4) Definition The canonical extensions % of the sheaf Ж (with V)
onto S is an extension corresponding to a trivialization of Ж by means of
the basis sj = s[Aj], j = 1, ...,//, where Ль ... , Ам € Я is a basis of
horizontal sections.
Thus, we have a trivialization
B.6.5) &S(H) = Я ®c ^Js -=¦ Jgf, Л •-» 5[Л].
Now we can interpret the limit Hodge fibration F's B.5.8) in the following
way.
F.6) Theorem Let (H, .F") be a variation of HS and let .9" - {F;} also
denote a filtration on Ж = JZ\S- ~ Я (g>c &s- Then the filtration .9'' is
extended to a filtration of i?, and F's is the filtration on the zero fibre
F\ = lim F\.
Proof. Defining Fms we first lifted the family of filtrations F) С Я, on U

70
II Limit MHSs
and, using a trivialization Hv = H Xs. U ~ H X U, we obtained a family
of filtrations on the space H, ф(и) = F'u = (/*)"' Ft, t = e(u), satisfying
the condition ф(и + 1) = Т~1ф(и).
In the same way, this trivialization permits us to consider any section
5 e Ж of the fibration H, after its lift 5 = s(e2-Ti") to a section of Ни, as
a section s € б u(H) of a constant fibration H X U: u*->su =
(/*)"'s(f) € H, where ~su = su(w) (considered as a horizontal section of
Hy, w € U) is such a horizontal section, which satisfies an initial condition
su(w)i^u = s(t) = s(e2™"). As for filtrations we have ~su+l = T~lsu.
In particular, if 5 = s[A] = tNA(t) = ebANu А(е2л™) е Ж is a section
constructed by a section A e H, then
3 (w\ — e2--ri№/ 4(ebn«\
Indeed, it is a horizontal section of Ни as a function in w, and it satisfies
the required initial condition. (This shows why 5M+i is transformed to
5u+i = T~lsu (and not to Tsu) - it is a dependence upon the initial
condition.)
The same formulae show that horizontal sections 5 = ^(e2311") of Hv
correspond to constant sections s = A = const of the fibration H X U.
Now we return to a family of filtrations F;, i.e. to a family of subspaces
Ff С H,. Here Ff is generated by base elements. Let a» be a base section
of.^P, a>(t) e Ff. Let us express it by a basis sy = s[Aj], j = 1, ... , (i,
of the canonical trivialization of.^ = M'y. Let со — J2%icjO)sj(t), where
Cj(i) € 6's'. Then the family of filtrations ф(и) = Fu on H consists of f_,
subspaces Fu, generated by base elements со = 1]у=1с,(е2лш)Eу)ц. The ~
'untwisted' family of filtrations Fu = xj){u) = e~2mN" ф(и) is generated
by base sections e~2ltiNu-cb = Y,%xcj(ebl'")Aj> where A} is a constant sec-
section of H X U. Now by definition Fps is generated by sections
lim^oZwLi^CO^y. and this means that F§ corresponds to a filtration
lim,_*o Ff on the zero fibre 3'o of the canonical extension, which is
generated by base elements lim^o'o = l»m,_oZ]y=icX0^> where Sj are
base vectors of a canonical trivialization of if?. ¦
2.7 The weight filtration of a nilpotent operator
B.7.1) Lemma-definition Let N: H —> H be a nilpotent operator on a
vector space H, Nk+l = 0. Then there exists a unique increasing filtration
W. = W(N) on H,
... с W-x с w0 с wx с ...,
2 The limit MHS according to Schmid
characterized by properties
71
is an
(i) N(Wt) С
(ii) Nr: Gr*'¦
This filtration W. = fF(A^) is called the weight filtration of a nilpotent
operator N (with centre 0 or with central index 0).
The filtration W(N) can be constructed by different means, for example,
by means of an iteration process. However, the most down-to-earth method
of construction of the filtration W{N) is the following.
Choose a Jordan basis щ, ... , Up of the operator N on H, i.e. such a
basis that wrt it the matrix of N has a Jordan normal form, i.e. consists of
blocks of the form
N =
/0
1
••• 0\ .
Vo ... 1 o/
, Nuq-x — u4, Nuq = 0
Number basis vectors correspond to each block in such a way that their
numbers are symmetric relative to 0 and that N decreases the index by 2
(i.e. N(ui) = M,-_2): .
if q = 2k+\ is odd,
then u2k, ¦ ¦ ¦ , u2, uo, m_2, • ¦ ¦ , И-2*;
if q = 2k is even,
then u2k-x и-ь Mb ... , и_B*-1).
In such way we attach a weight to each base vector - this is its number in
the numeration, and we have N(ui) — м,-_2. Let Wk = & subspace generated
by vectors of weight =s k. Obviously, properties (i) and (ii) are then
fulfilled. ¦
We can visualize the filtration W, by means of the following schematic
diagram. In this a base vector is represented by a square with a weight
written in it. Base vectors of any block are set in a line at a distance from a
fixed axis which is equal to their weight:

72
IILimitMHSs
3
2
1
N
1
0
N
0
-1
iV
— 1
-2
B.7.2) Definition A weight filtration of a nilpotent operator N with centre
n (or with central index n) is a filtration "ff. obtained from W. — W{N)
by a shift of numeration
"W. = W\-n\ i.e. "Wk = fF_n+*.
For the filtration " W. property (ii) can be written as:
B.7.3) Nr: Gr"n\r-^> Gr"nZr is an isomorphism for all r.
Let us make a simple, but very useful, note needed in the next section.
B.7.4) Let H = H/KsrN, and let N be an operator induced by N on H.
Denote by W a weight filtration (with centre 0) of the operator N on H,
and by f a filtration induced by the filtration " W on H.
B.7.5) Proposition
»W= n+1W,
i.e. a filtration induced on H by the weight filtration on H with centre и
coincides with the weight filtration of the operator N on H with centre
л+1.
Proof This is shown in the following diagram
3 The limit MHS according to Steenbrink
73
-1
—
-2
-3
and so on. Here KerN is generated by base vectors represented by the
unshaded squares, and H by shaded squares. ¦
2.8 Schmid's theorem
B.8.1) Theorem Let (H, J<~) be a polarized variation of an HS of weight
n over a punctured disk, // be a space of horizontal (multivalued) sections
of H, and ГЬе the monodromy operator. Let F's be a limit Hodge filtration
introduced in B.5.8), W. be a weight filtration of the nilpotent operator
N = — (l/2jri)Iog Tu (the logarithm of the unipotent part of the mono-
monodromy) with centre n. Then the filtrations (W., F's) determine an MHS on
H, which is called the limit MHS of the given variation of HS. With respect
to this MHS the operator 7У: H —> H is a morphism of the MHS of type
(-1, -1), i.e. NF§ с Fps'\ NWk с W^2.
The proof of this theorem follows from the Schmid's SZ,2-orbit theorem
[Sm].
3 The limit MHS according to Steenbrink
In this chapter we are mainly interested in the problem of introducing the
MHS on the cohomology of a canonical fibre ^(X^) = H of the Milnor
fibration /: X -* S associated with an isolated singularity /:
(C"+1, 0) -»¦ (C, 0). This was first done by Steenbrink [S3] by using an
embedding of / into a family of projective hypersurface тс: Y —> S and

74
II Limit MHSs
then resolving the singularities of this family. He used the limit MHS of a
variation of an HS constructed by him in the geometric situation of a
family of projective varieties. In §2 we saw that according to Schmid a
limit HS of a variation of the HS (Ж, V, .У') over a punctured disk
naturally appears on the zero fibre of the canonical extension S? of the
sheaf Jig to the point 0 € S. In the geometric situation in order to extend
the sheaf 30 and filtrations .9'' over S Steenbrink used sheaves of relative
differential forms with logarithmic poles. The filtrations F~ and W. of the
limit MHS are introduced in the spirit of Deligne by means of filtrations on
complexes of sheaves whose hypercohomology coincides with H. Our goal
is to introduce the MHS on the vanishing cohomology without using an
embedding in a projective family and resolution of singularities (this is
Varchenko's idea). Therefore here we'll give only a very schematic account
of Steenbrink's construction, without going into technical details.
3.1 The limit MHS for projective families: the case of unipotent
monodromy
Let ж: Y —> S be a family of projective varieties (induced by a projection
У С PN X S -»¦ S), which is smooth over a punctured disk S' = S\{0},
and let Уо = лг'(О) be a degenerate fibre. Resolving singularities of Yq,
we can assume Уо to be a divisor with normal crossings. First we'll
consider the case in which Yq is a reduced divisor.
C.1.1) The family ж defines a variation of HS (.^f, V, i*"') on the sheaf
Зё = R"n*Cy ®c $'s' — Wn*Q'r,s,, where ж: У —> S' is a restriction
of ж onto У = У\Уо- The connection V is defined by a local system
H = R"n*CY', and the Hodge filtration &"', as is well known, corresponds
to a stupid filtration on the De Rham complex Qyy5., which is a resolution
of the sheaf n~l6>'s' (the relative Poincare lemma), .9~p = M"n*FpQr,s,,
where Fp = oSp:...-+ 0 -»¦ Q',,.,
p+1
C.1.2) Steenbrink proves that a sheaf.Ж = R"jr*Qr/s(log 70) is locally
free on S and, consequently, is a natural extension of 3i? over S. The
connection V is extended to a connection on Ж with a logarithmic pole at
0 € S, V: ~Ш -+ Q^Gog 0) ® ~M, (i.e. Ж is a saturated lattice in the sense
of A.7.7.2)). The eigenvalues a of the connection residue Res0 V on 3%
satisfy the condition 0 =? a < I (N. Katz). (In our case, when Уо is reduced,
the monodromy T = Tu is unipotent and hence a € Z since A = e2ma = I,
and, consequently, 3% coincides with the canonical extension S? F.3.5) for
3 The limit MHS according to Steenbrink 75
which -I< a =s 0.) Sheaves Wp = Iflj/', where j: S' -* S, ex-
extending the filtration .2~' over S according to Schmid, coincide in our
geometric case with sheaves jR"^FpQ'r /5(log Уо).
C.1.3) The limit MHS in a natural way 'lives' in the vector space
H — Н1"(Уо, Q'jy5(log Уо) ® @y,) isomorphic to the zero fibre of the sheaf
Ж. The Hodge filtration F- on H is induced by the filtration ~W- on Ж. It
corresponds to a stupid filtration on Q'y/,s(log Уо)® &y0. The weight
filtration W. on H can be defined in the same way as in §2 (the weight
filtration of the logarithm of unipotent part of the monodromy). Steenbrink
proceeds in the following way. He defines a complex A\ quasiisomorphic
to the complex Q"y/5(log Уо) ® (9Yo, and defines two filtrations W. and F~
on it. Then he proves that (A\ W, F) is a cohomological mixed Hodge
complex in the sense of Deligne, which induces an MHS on
H = Н"(У0, А-) coinciding with Schmid's limit MHS.
3.2 The limit MHS for projective families: the general case
Let ж: Y —» S be a family of projective varieties and лг'@)
= Уо = moEo + ... + m/cEk = E be a divisor with normal crossings. Let
us perform a base change S—*S,t = tm, where m = Icm (mo, ..., mk). Let
У be a normalization of У X5 S, and я: Y —> S and (p: Y —» У be natural
mappings:
C.2.1)
=Y0CY
Yoo =
U.
5'
Let Z),- = ^-'(?/), j = Q,...,k. Denote by У^ = У X5. J7 = У' X-s, U
the canonical ('general') fibre of the family ж. Here U is the upper half-
plane, and e = pe: U -* S',t— e2", is the universal covering.
C.2.2) Lemma [S3] У is a ^-manifold, i.e. it has at most finite quotient
singularities, and тг"'(О) = D = Д + ... + Dk is a reduced divisor with
^normal crossings on У. ¦
C.2.3) Steenbrink extends the theory of the previous item [SI] to V-
manifolds [S3]. The role of sheaves of differential forms Q? on a ^manifold
У is played by sheaves (axQp)G, where (locally) У = Z/G, a: Z-*Y, i.e.
sheaves of invariant differential forms on a smooth manifold Z covering
У. Analogously one defines sheaves Q?(Iog Уо), Щ,-$> Щ

76
II Limit MHSs
Projective ^manifolds behave in all respects as non-singular protective
varieties. In spite of the fact that the ^have weak singularities, the MHS on
H"(Y,, C) is pure of weight n, there is a Lefshetz decomposition and so on.
C.2.4) The sheaf .Я? = Mnn*Qy-s(logYo) is locally free and gives a
natural extension of the sheaf Ж = р1'Ж = R"Jr*Qp,,s, to the point
0_e S. The limit MHS in a natural way 'lives' in the zero fibre of this sheaf
Jg?o = H"(Y0, Gf/sQ-og Yo) ®r -r f y0). In the same way as in C.1.3) one
defines a complex A' quasiisomorphic to the complex Qj,,^(log Yo)®r-Y
б Y and also two filiations W. and F\ etc. We thus obtain an MHS on the
space W(Y0, A').
A choice of local coordinates t and r, xm = t, defines the isomophisms
C.2.5) H"(?o, Qp/S(log
r у #y0)
, C) tr
Yo)
This enables us to carry the MHS to the cohomology of the canonical fibre
Я"(Гоо, С) of the family л.
Denote by T and t = Tm the monodromy operators on #"(Уос, С)
corresponding to generators of Я[E") and Jt\(S'). Then the monodromy
f = fu is unipotent. Denote the logarithm of the unipotent part of the
monodromy by
C.2.6)
m
m-
C.2.7) Theorem [S3] There exists an MHS on H"(YX) such that
(i) N: Я'Ч^оо) -> H^Yoo) is an endomorphism of the MHS (Я
W., F-) of type (-1,-1), i.e. N(Wk) С Wk_2, N(FP) С F"'1.
(ii) For all г Э= О,
is a MHS 45ф
(iii) The semisimple part Ts of the monodromy is an MHS isomorphism.
¦
Let us consider Hodge numbers of the MHS on #"(Уоо)
C.2.8) hM = dim GrpFGrwp+qHn{Yoa).
By the monodromy theorem (A. Landman and others, cf §1.9) we have
Nn+l = 0 and, consequently, from the definition of W. we have
3 The limit MHS according to Steenbrink 11
0 С Wo С Wx С ... С Wlu-\ С Wln = H"(YX).
For the limit Hodge filtration F- as well as for the Hodge filtration on the
cohomology of a non-singular projective variety of dimension n we have
OCF'C F"-x с ... С F° = H"(YX).
From this and by the definition of Hodge numbers it follows that:
C.2.9) hp'q not equal to zero are contained in a quadrate [0, n] X [0, и];
C.2.10) hM = hq'p (this is a symmetry relative to the diagonal - the
bisector of the coordinate angle);
C.2.11) hp<4 = h"-^"-p (this is a symmetry relative to the second diag-
diagonal); it follows from the second part of theorem C.2.7) because there is
an isomorphism
-: GrpGrJ+q - Gr"pGrl_p_r
3.3 Brieskorn construction
Let /: (C+1, 0) -> (C, 0) be an isolated singularity of a holomorphic
function. Brieskorn [Br] proposed the following construction of an embed-
embedding of/to a family of projective varieties я: Y —> S. The singularity /
being isolated, we can assume after a change of coordinates that / is a
polynomial. If we add a homogeneous polynomial h(x0,... , xn) of suffi-
sufficiently large degree d to / then, on one hand, we obtain a singularity
analytically isomorphic to the initial singularity at 0 e C"+1 because /is
isolated. On the other hand, if d is sufficiently large and h is chosen
sufficiently general, then the hypersurface in P"+1 with the equation
f(x) = 0 will have singularities only at the point 0 € C"+1 (the Bertini
theorem). Let us consider a family of hypersurfaces Y с P"+1 X С with
affine equations f(x0, ... , х„) = t or with projective equations
Xn+U
~» • • • . ~ 1 ~ tXn+\ - »•
X Л J
Let я: Y —> С be induced by projection. Then я has at most a finite
number of degenerate fibres Y,, and, choosing a disk S Э 0 of sufficiently
small radius, we can assume that я has a single degenerate fibre To over S.
We obtain the following picture

i \
li\
78
C-3.1)
jn+l
II Limit MHSs
D7 D }.
s J'
where я is a projective morphism, smooth over 5", which has in the fibre
jf"'(O) = To a single singular point 0 6 70 analytically equivalent to the
germ of hypersurface Xo: Дх) = 0, and X = 7h В is the intersection with
a ball В of small radius centred at 0 6 Cn+1, and /: X -* S is the Milnor
fibration of the singularity /
3.4 Limit MHS on a vanishing cohomology
Let /: (C+1, 0)-> (C, 0) be an isolated singularity. Embed it in a
projective family by means of the Brieskorn construction C.3.1). Let
a: Y —> 7 be a resolution of the singularity of the map я, i.e.
Y\o~]@) i> 7\0 is an isomorphism, and the zero fibre of the map
л = яо: Y —* S is a divisor with normal crossings лг"'@) = Yo = Eq
+ m\Ex + ... + mkEk. Then я: Eo —> 70 is a good resolution of the
singularity (To, 0) with exceptional divisor С = \J Cj, where
Cj = Eo П Ej, j = 1, ... , k. We find ourselves with a projective family я:
Y —* S, in the cohomology of the canonical fibre Yx = Y^ of which the
MHS was denned in C.2.7),
C.4.1)
yZ-yz
~я\1я
S
Outside the ball В э 0 the map я: Y\B = Y\X -»5isa locally trivial
C°°-fibration. The fibre Xo =/"'@) is a cone over К = Xo П OB with a
vertex at the point 0. Therefore the fibre 70 has the homotopy type of a
quotient space 7,/X, = Y,/X,, t ф 0. Thus Я'G0) ~ H%Yt, X,). The
CA exact cohomology sequence of the pair (У,, X,) gives an exact sequence:
C.4.2)
0 -» Hn(Yo) -
Hn(Yt)
Я"+1G0)
0
because Xt has the homotopy type of a bouquet of [i spheres S" and
Я'^Х,) = 0 for i: ^ 0 and и. Here /* is induced by the embedding
i: XXL Y, and r* is induced by the retraction r: 7 —> 70. Homomorph-
isms in this exact sequence are equivariant wrt the monodromy T because
one may take the geometric monodromy h,: Y, —> Yt to be the identity
outside X,.
3 The limit MHS according to Steenbrink
79
Let Уоо = Y Xs. U = Y X-s, U and X^ = X X5- I/ = X X-s. U be the
canonical fibres of the maps л: (which is the same as of maps "n or я) and /
The exact sequence C.4.2) can be rewritten in the form
C.4.3)
С »(Гоо) ^ Д"(^оо) - Яя+1G0) - Я-'+ЧУо
0 - Я"Gо)
0.
All the terms of this sequence except Я"(АгО0) already carry a MHS
structure. Уо is a projective variety with a single singular point 0 6 70 and
a complement У<Д0 ~ E$\C, where С is a divisor with normal crossings.
In the spirit of Deligne Steenbrink [S3] constructs an exact sequence
0 —> A(Y0) —> A' —у А\ХЖ) —> 0 of cohomological mixed Hodge com-
complexes, the hypercohomologies of which are Я?G0), H9(Yoo) and
НЧ(ХЖ). This implies the following theorem:
C.4.4) Theorem The vanishing cohomology Я"(Аг0О) carries an MHS
such that the exact sequence C.4.3) is an exact sequence of the MHS. The
MHS on Я"(АГОО) is invariant wrt the semisimple part Ts of the mono-
monodromy. ¦
C.4.5) One can show [S3] that the MHS on Ял+1G0) is pure of weight
n+1.
5.5 The weight filtration on H"(Xao). Symmetry of Hodge numbers
Scherk and Steenbrink [Sc-S] showed that if in the Brieskorn construction
we choose deg/ = d sufficiently large, then i* is an epimorphism, i.e. we
have an exact MHS sequence
C.5.1) 0 -> ЯЛ(УО) -> Я^Уоо) -U Л"(ЛГоо) -> 0,
and we can consider the MHS on H"(Xao) to be a quotient of the MHS on
C.5.2) The weight filtration Won #"(*«,) as well as that on Я"(Уоо) is
the weight filtration of the nilpotent operator N logarithm of the unipotent
part of monodromy Ta.
The local invariant cycle theorem [SI] asserts that the MHS sequence
C.5.3)
Яя(У0)^Ял(Уоо)Дяя(Уоо)
is exact (it is also a part of the Clemens-Schmid exact sequence). There-
Therefore the image of r* coincides with KeriV = Кег(Ги — id) =

80 II Limit MHSs
KerNП Hn(Yx)x (and KerNП Н"(Ух)ф1 = 0). Here H"(YX) =
©л#"(Уэс)л is a root decomposition corresponding to the eigenvalues Я of
the monodromy T, H"(Yx)^i = ®д^ Я"(Гэс)а. Since the MHS is invariant
wrt the semisimple part of monodromy Ts, we have from C.5.1):
C.5.4) A) Нп(Хк)ф1 = Hn(Yx)^i and the weight filtration W. on
#"(^00M*1 as wel1 as on H"(Y°c)is the weight filtration of the operator N
with centre n;
B) H"(Xx)i = #"(roc)i/Ker(W)i and the weight filtration W. on
Hn{XO0)\, as was noted in B.7.5), is the weight filtration of the operator N
with centre и + 1 (!).
Now let hp>i be Hodge numbers of the MHS on H"(XX)
C.5.5) Л"« = dim Gr'pGr^inXn).
We have a decomposition into a sum /гр-« = Yli-K'" = АГ* + *?'* over
Hodge numbers corresponding to different eigenvalues of the monodromy
T. As with #"(Уоо) the Hodge numbers h™ ^ 0 are contained in a
quadrate [0, n] X [0, и].
There are two types of symmetry for Hodge numbers, and for the second
type the symmetries for Я = 1 and Я ^ 1 are different:
(i) Since the Hodge filtration Fm induces on Gr J+<? an HS of weight
p + q, we have
C.5.6) Ал« = h"'p,
or in more detail h%'4 = h-'p,
иРЛ _ ifl'P UPA — hq'p-
 —  . "/1 — "/1 >
in the matrix (/гр>?) this is a symmetry relative to the accessory
diagonal,
(ii) From the second part of theorem C.7) we have
NP+q-n. Gr^H^X^ - Gr^^H^X^u
and (taking into account the shift of the central index)
We obtain isomorphisms
NP+4-n. OrpOr^If
and
and, consequently,
n +
C.
C.
я,
5.7)
5.8)
&."
О
\
3 The limit MHS according to Steenbrink
_ i,n-q,n-p
— пф\
81
1 P
In the case of A^f this is a symmetry relative to the main diagonal of
the matrix (hp'q), 0 =s p, q =e n, and in the case of hp'q it is a
symmetry relative to the main diagonal of a (n + 1) X (и + l)-matrix.
It follows that F0#"(^oo)i = ^#"(^00I = ° and we obtain the
following more precise version of the monodromy theorem A.9.1.1).
C.5.9) Theorem The Jordan blocks of the monodromy Ton H"(Xoo) are
of size at most и + 1. The Jordan blocks for eigenvalue Я = 1 of Гаге of
size at most п. Ш
It is clear that the following conditions are equivalent: (i) there is a
Jordan block of size n + 1 (and therefore necessarily with Я Ф 1); (ii)
Gr2nHn(Xx)^i ^ 0 (correspondingly: (i) there is a Jordan block of size и
with Я = 1; (ii) Gr^H^X^i ^ 0).
Van Doom and Steenbrink [DS] obtained the following:
C.5.10) Supplement to the monodromy theorem:
Gr?nH\Xx)^ ± 0 => Gr*nH*(X^x ± 0,
i.e. if the monodromy T on Ял(АГОо) has a Jordan block of size и + 1

82
11 Limit MHSs
(necessary for an eigenvalue Я Ф 1), then Talso has a Jordan block of size
n for the eigenvalue Я = 1.
C.5.11) This supplement is a generalization of the following result of Le
Dung Trang [L]: The monodromy Г of an irreducible one-dimensional
singularity/: (C2, 0) —> (C, 0) is of finite order.
Indeed, the finiteness of order of T is equivalent to the absence of two-
dimensional Jordan blocks. However, if there is such a block (of size
и+1=2), then T has the eigenvalue Я = 1. And as is known, f(x, y) is
irreducible if and only if T has Я = 1 as an eigenvalue.
4 Hodge theory of a smooth hypersurface according to Griffiths-
Deligne
Our main subject of study is the Milnor fibration /: X —» S of an isolated
singularity/: (C"+1, 0) —> (C, 0) and the linearization associated with it -
a local system H-> S', in which H = {J teS-H"(Xt, С) = Д7*Сд-. is the
cohomological fibration. Fibres of/, X, =f~l(t), as well as fibres of its
projective embedding n: 7 —> S, Yt = Jt~l(t), are non-singular (for t Ф 0)
hypersurfaces in a complex manifold X (correspondingly, in Y), and fibres
H, = H"(Xt, C) are a cohomology of such hypersurfaces. Therefore
before we proceed, we consider how to calculate the cohomology of a
hypersurface or its complement by means of differential forms on the
ambient manifold with poles on this hypersurface and how the Hodge
filtration is connected with the pole order filtration. The answer to this
problem is supplied by Griffiths-Deligne theory ([Gr] or in the MHS
context [D3]). Temporarily we change our notation.
4.1 The Gysin exact sequence
Let ХЪе a complex manifold, dimX = n + 1, and Y С X be a hypersur-
hypersurface defined by the (local) equation s = 0, U = X\Y, and j: U с X be an
embedding. The cohomologies of X, 7 and U are connected by an exact
sequence
Hk~2(Y, C) -> Hk(X, C) -> H\U, C) -> Hk~\Y)
D.1.1)
which is called the Gysin sequence.
From the topological point of view this sequence may be considered in
one of the following three ways:
4 Hodge theory of a smooth hypersurface 83
A) As an exact sequence obtained from the degenerate Leray spectral
sequence for the embedding j: U —* X
D.1.2) E{« = H?(X, RqUCu) => H"+"(U, Си).
In our case j^Cv = Cx, Rlj*Cu = Cr (locally), and Rqj*Cu = 0
for q > 1 because for a ball neighborhood V с X at a point у е Y the
inverse image j~l(V)= V\Y has the homotopy type of a circle S1.
B) As an exact cohomology sequence of a pair (X, U) or an exact local
cohomology sequence -^
D.1.3)
Hkr(X) -» Hk(X) -> Hk(U)
where HkY(X) = Hk(X, U). The cohomology HkY(X, C) may be con-
considered as derived functors of the functor H°r(X, •) of sections with a
support in 7 of a sheaf on X. From a spectral sequence connecting
sheaves of local cohomology .%?Чу(.9~) with local cohomology
D.1.4) E{* - ЩХ, ЗРГ(.Г)) =»• HPY+\X, .Г).
In our case for & = Cx we have .Щ{Х, Сх) = 0 for q ф 2 and
.%?2Y(X, С) ~ Су (locally) because for a neighborhood Xof a point у
we have H"Y(X, Cx) = НЦХ, X\Y) = Я'02, ?>2\{0}), where D1
is a 2-disk. The terms E%'q vanish except when q = 2 and this gives an
identification
D.1.5) H\{X, Cx) = Нк~\х, Cr) = Hk-2(Y, CY).
C) As an exact Gysin sequence associated with a spectral sequence of a
fibration with fibre Sl. One needs to consider a tubular neighborhood
of 7in X.
D.1.6) Remark A non-singular hypersurface Y С X is a particular case of
a divisor with normal crossings in X considered by Deligne. The Gysin
sequence D.1.1) is a particular case of a Leray spectral sequence for an
embedding/.
4.2 Hodge theory for a complement U = X\Y. Hodge filtration and pole
order filtration
Deligne considered a case in which 7 is a divisor with normal crossings.
Any non-singular quasiprojective variety U may be presented in the form
X\Y, where Xis a non-singular projective variety and 7is a divisor with
normal crossings. This enables us to introduce an MHS on H-(U).
D.2.1) By the Poincare lemma we have a resolution 0 —* С у —> Qy and

84
II Limit MHSs
hence H'(U, C) = Hl"(Qy). The embedding j is a Stein morphism
(because У is a hypersurface) and hence Qv is a j* -acyclic resolution,
№j*-7~ = 0 for q > 0 for coherent sheaves, and, consequently,
#•(?/, С) = №(/*^).
Denote by ?2}^-(*У) С j* Q.'v a subcomplex of sheaves of meromorphic
differential forms on ^fwith poles along У Let Q>(log У) be a complex of
sheaves of differential forms on ЛТ with logarithmic poles along Y. If Y is a
divisor with normal crossings, then these three complexes
are quasiisomorphic, i.e. their cohomology sheaves are isomorphic. Conse-
Consequently,
D.2.2) #(?/, С) ~ H-(Q^(log У)).
* If У С X is a non-singular hypersurface, then there is an exact sequence
of complexes of sheaves on X
D.2.3)
0
Q>(log У)
0,
where R is the Poincare residue which takes the form со =
ip Л ds/s 6 Q^-(log Y) to the form V|y € Я*. The exact hypercohomol-
ogy sequence associated with the sequence D.2.3) is nothing but the exact
Gysin sequence D.1.1).
D.2.4) Remark The inclusion О С Qx = Wo С Q>(log У) = W\ in the
case a non-singular hypersurface У is a particular case of the weight
filtration W, on Q>(log У).
D.2.5) Now we come to the Hodge filtration F- on #(?/)¦ As is known,
under the isomorphism H'{U, C) ~ H'(Qx{\og У)), filtration F' on
#(?/, C) corresponds to a stupid filtration стЭ/, on the complex
Q^log У), where for a complex AT" by definition o^pK' is a complex
0 —». -+ 0 —» ^ —> Kp+l —»....
Now we define a />o/e order filtration P' on the complex Q^+y). First
we define an increasing filtration P. on the zero term of this complex
D.2.6) 0 С &X{Y) с
I
С ... С
+ 1)У) С ...
II II II II
P-\ Po . Pi Pk&x(*Y).
Let P' be the corresponding decreasing filtration, P* = P_*,
D.2.7) Р*^>(*У) = ^((-* + 1)У), Рк = 0 for к > 0.
4 Hodge theory of a smooth hypersurface 85
A differentiation leads to an increase in the pole order of 1. Hence, to
spread this filtration to Q^(* У) respecting d, it is natural to set
D.2.8) Р*Я?(*У) = Pk-?6X{*Y) ®QX = Qx((p - k+ \)Y).
We obtain a filtration on QX(*Y) by subcomplexes P*
D.2.9)
и
и
и
P1:
и
P2:
и
и
Р*:
и
и
и
и
0
0
0
и
Й^(ЗУ)
и
It is obvious that on the subcomplex Q^log У) с ?2>(*У) the filtration
coincides with the stupid filtration o^p,
D.2.10) PP
D.2.11) Theorem The inclusion
filtered quasiisomorphism.
(log У),
(?2>(*У), Р) is a
4.3 De Rham complex of the sheaf B\y\x and the cohomology of a
hypersurface Y
We may use the inclusion Q^log У) С QX(*Y) and embed the exact
sequence D.2.3) into a commutative diagram with exact lines
D.3.1) 0^Q> -» Qx(logy) -Д./*Яу[-1]->0
II *П |a
O^Q-^ Q>(*y) -^DR(B[Y]X)^0,
where by definition
D.3.2) DR(B[y]x) = ^V(*y)/Qv = ^Vig) (d?x(*Y)/&x).
Df /«
The quotient complex DR{B[Y]X) is the De Rham complex of the sheaf of
the principle parts of meromorphic functions with poles along У
D.3.3)

86
II Limit MHSs
The mapping a is defined by commutativity: if со € Qy is a p-form on Y
(locally), then first we lift it to Qx(\og Y) getting a (p + l)-form a> Л ds/s,
where a> is an extension of a> to X, and then we consider a class of the form
cbAds/se QP+[(*Y)modQx+] as an element а(ш) € DRP+i(B[Y]X).
The pole order filtration on Qx(* Y) induces a pole order filtration P~ on
DR(B[Y]x)- Since the homomorphism b is a filtered quasiisomorphism
D.2.11), we obtain the following theorem:
D.3.4) Theorem The homomorphism
a: (/*Qy[-l], a») -> (DRB[Y]X, F)
is a filtered quasiisomorphism. ¦
D.3.5) Corollary
H\Y, С) ~ Hk+\X, DR-(B[Y]X),
and under this isomorphism the filtration Fp on Hk(Y, C) is induced by
the filtration Pp+X on the complex DR(B[Y]X).
4.4 The case of a smooth hypersurface Y in a projective space
A. — Vr
On the cohomology of the complement U = P"+1\Y
Hd(U, C) = Ud(X, Q>(log Y)) = UdX, QX(*Y))
the Hodge filtration Fp is induced by the stupid filtration ст3 on Q> (log Y)
or by the pole order filtration P~ on QX(*Y), i.e. subspaces FpHd{U) are
images of spaces Ud(X, PPQ.X(*Y), where
D.4.1) P"QX(*Y): 0-»...-»0-
(QjG) stands on the pth place).
By Bott's theorem
D.4.2) H\X, Qx(kY) = 0 for к > 1, i 3= 1, p. ** 0.
This means that the complex PPQX(*Y) is acyclic wrt the functor of
sections, and hence to calculate the hypercohomology we may take the
corresponding complex of global sections
D.4.3) Hd(X, PPQ^Y) = Hd(T(X, F'Q^Y)).
It follows that the cohomology classes с € FpHd(U) are represented by
4 Hodge theory of a smooth hypersurface 87
closed rf-forms a € H°(X, Qx((d ~p+ 1O) with poles of order
=e d-p + l.
Let us consider the middle cohomology Hn(Y, C) of a hypersurface Y,
dim Y = n. In the Gysin sequence D.1.1)
...->л-Чг)
-* H"+[(Pn+l) -> Hn+\U) А Я"(У) Л Я"+2(Р"+1) -* • • •
the homomorphisms g sends с to /г Л с where h is the class of the
hyperplane section. Denote by Pk(Y, С) С Hk(Y, C) the primitive coho-
cohomology of Y Then P"(Y, C) = Kerg and we get an isomorphism
Hn+\U)^i P"(Y). From the above it follows that the Hodge filtration on
P"{Y) is of the following sort:
D.4.4)
Hn+\U) = JW?\(Y) D
P"(Y) = F°P"(Y) D
where
D.4.5) Л-^-Н (ЦР '
О
F»P"(Y)D ... D
4.5 Generalization to the case of a hypersurface with singularities
D.5.1) Let Xbe a complex manifold, dimX = n + 1, and let Y С X be a
hypersurface, possibly with singularities, U = X\ Y. In the same way as in
D.2.1) we have H-(U, С) ~ U-(&X(*Y)). There is a pole order filtration
P~ D.2.9) on the complex QX(*Y), which induces a pole order filtration
P' on the cohomology of the complement H'(U, C). We want to compare
this with the Hodge filtration F' from MHS theory.
In the global situation of D.2.11) and D.4.1) we make the following
generalization:
D.5.2) Theorem [D-Di] If Y is an hypersurface in a projective space
X — P"+1 or even in a weighted projective space P = P(w0, ... , wn+1)
and F- is the Hodge filtration on H'(U, C) ([D3]) then for all к and m we
have
PkHm(U) Э FkHm(U). Ш
If a hypersurface Y is (quasi)smooth, then P- — F\ In general, this is not
the case (see, for example, (VI.1.33) in [Di]).

88
II Limit MHSs
The position is not quite the same in the local case as it is in the global
one:
D.5.3) Let /: X —> S be the Milnor fibration of a hypersurface singularity
and Y = Xo - /~'(°) be the singular fibre. In this case U = X\Y plays
the role of a knot of the singularity. Usually the intersection К =
/-'@) П S2"+l with a sphere S2n+l = dB2n is called a knot of a singul-
singularity. The pair (X, Y) is homeomorphic to the cone on (S2n+l, K) with the
vertex xo=OeI Hence, Y\xq and X\ Y are homotopy equivalent to К
and S2n+1 \K. Applying the Alexander duality to^c52"+1,we obtain
Thus from the homological point of view we may consider U = X\ 7 as a
knot of the singularity.
D.5.4) Embedding 7in a projective variety as in C.3.1) and applying the
excision axiom, we may assume that 7 is a projective variety. In this case
there is an MHS on H(Y\x0) ([D2]). Let F~ be the Hodge filtration of this
MHS. The filtrations F~ and Pm are connected by the following relation:
D.5.5) Theorem ([Kal, Di2]) If f X -> S is the Milnor fibration of an
isolated singularity /: (C"+1, 0) ->(C, 0) and F- and P- are the Hodge
and the pole order filtrations on H(X\Xo, C), then
(i) for all к
(ii) for all*
D FkHn+l(X\X0);
PkH"(X\X0) С FkHn(X\X0).
5 The Gauss-Manin system of an isolated singularity
We now study the Gauss-Manin connection on the sheaf.Ж = #<Эс <^V>
H= Л"/*Сл" = U teS'H"(Xt, С), associated with the Milnor fibration /:
X -> S of an isolated singularity f. (C"+1, 0) -> (C, 0). A fibre of the
sheaf Ж is the cohomology H"(X,) of a non-singular hypersurface
X, с X. In the previous section we saw how one can study the cohomology
of the hypersurface by means of differential forms on X with poles along
X,, Hn(Xt, C) ~ U"+l(X, DR(B[Xi]X)). Now we want to extend this to
the relative case - to a family of hypersurfaces X,, t e S", and then to
apply this to get a natural extension of the sheaf .Ж onto the whole disk S.
5 The Gauss-Manin system of an isolated singularity 89
In this section we try to explain of the appearance of the differential system
Л?х in [Sc-S], without using the language of the theory of Z)-modules.
5.1 Hodge theory of a smooth hypersurface in the relative case
Let /: X —> S be a morphism of a complex manifold X, длтХ = n + 1,
to a curve S. For any t e S we have an embedding of a hypersurface
f~](t) = Xt С X. Varying t we glue the fibres X, together into a graph Г
in the manifold Z = X X S. A graph Г С X X S is a smooth hypersurface
in Z defined by the equation f(x) - t = 0. The mapping / factors to a
composition of an embedding i: X С Z onto the graph and a projection p
E.1.1)
f
The intersection of the fibre p '(/) = IX{;} with the hypersurface Г is
the fibre X, = f~l(t), and the pair X, с X is isomorphic to the pair
Г, с Z,.
The sheaf v
E.1.2) Blf]xxs = %]Z = б z(*ry?z
of principal parts of the sheaf of meromorphic functions <^z(*r)
= <^'z[l/LA*) - t]] with poles along Г: f(x) - t = 0 on Z is a relative
analog of the sheaf B[X,]x in D.3.3). Consider the relative De Rham
complex of this sheaf
E.1.3) DRz/s(B[r]Z) = Qz/s ®r- z %]z = Q-z/5(*r)/Q-z/s.
E.1.4) Proposition If /: X —> S is smooth, then
Rkf*Cx ® &s - Uk+lp*(DR-z/s(B[r]Z))
(this is a relative variant of corollary D.3.5)).
Indeed, if / is smooth, then the sheaves Q?,s are locally free and we
have the commutative diagram
0 -» Q-z/s -> Qz/S(logr)
|
0
la
0 -» Q-z/s -> Q'z/S(*r) -» DRz/siB^z) -» 0
generalizing diagram D.3.1) to the relative case. In this diagram a is a
quasiisomorphism. From this and from the relative Poincare lemma (for

90
II Limit MHSs
smooth /there is a resolution 0 —> / X6> s —* &x/s °f me sheaf/"
we obtain
Zx®cs6's =
5.2 ГАе Gauss-Manin differential system
In the general case, when /is not necessarily smooth, we define a complex
E.2.1) \&x = №p*(DR-z/s(B[T]Z))[n + 1].
E.2.2)
Its cohomology sheaves
f*
^ = Ukp*(DR-z/s(Bir]z))[n + 1]
= n»+l+kp*(DRz/s(Blr]z))
are called the (&th) Gauss-Manin differential systems of a morphism f
In particular, in the case of isolated singularities we are interested in the
sheaf
E.2.3)
= U"+l
U"+lp*(DRz/s(B[T]z)),
where /: X —> S is the Milnor fibration. This sheaf is called the Gauss—
Manin differential system of the singularity f
Since the morphism f\xr- X' —+ S' is smooth, it follows from proposi-
proposition E.1.4) that
E.2.4) 3%x\s' = Rnf*CX' ®c @s = Ж,
i.e. the sheaf *%?x is a natural extension of the sheaf Ж to the point 0 6 S.
E.2.5) Proposition If/is a Steinian morphism, for example, if/is the
Milnor fibration of a singularity, then
+1
where $fk(.) denotes the ?th cohomology of a complex of sheaves.
A standard deduction of formula E.2.5) from definition E.2.3) is the
following. Since / = ~p is Stein, we obtain by Cartan's theorem В that the
complex DR'z,s(B[r\Z) is acyclic relative the direct image functor. This
enables us to calculate hypercohomology by applying p* directly to this
complex. However, here the sheaves Q4Z,S(*T)/Q4Z,S are only quasicoher-
ent (not coherent) «^-modules, and Cartan's theorem В does not hold in
5 The Gauss-Manin system of an isolated singularity
91
general for quasicoherent sheaves, but one can overcome this difficulty
[Sc-S].
5.3 Interpretation of the complex DRz,s(B[r\z) in terms of the
morphism f:X—+S
The De Rham complex DRz/s(B[r]Z) = QZ/S(*T)/QZ/S E.1.3) is con-
concentrated on the graph Г because QZ/S(*T) and ?2Z/S coincide outside Г.
By means of the isomorphism i: X -^ Г С Z = X X S we carry to X the
complex DRz,s{B[Y]z) and all its natural structures (i.e. differential d, the
structure of the ?>s-module, the pole order filtration inducing an HS on
fibres of/in the case of a projective morphism).
Obviously, we have an isomorphism
E3Л) Q>[/(^] ^ rlQZ/5(*n/?2z/5-
Indeed, if a q-fona co/(f(x) — t)k represents an element of the right-hand
side, where cd is a «7-form with coefficients depending on x and t, then
changing t by f(x) — (f(x) — t) and redecomposing coefficients we can
assume that cd G Qx.
Complex E.3.1) naturally is a complex of ZVmodules, where on a form
cd/(J{x) — t)k, cd G Q^, the operator d, acts as a differentiation wrt the
parameter t.
E.3.2) d,
This implies that
E.3.3) d\
= k
if(x) - i
UCD
(here the brackets [ ] mean the class of an element in the quotient complex
in the right-hand side of formula E.3.1)).
Formula E.3.3) shows that we can consider the complex
Qx[l/(f(x) — t)], elements of which are polynomials in l/(f(x) ~ t), as a
complex Q^[Z)], elements of which are polynomials in a formal indetermi-
indeterminate D interpreted as the differentiation operator dt, i.e. there is an
isomorphism
E.3.4)
- t
CD
1У ^ d\
CD
f{x) - t
V.CD
(fix) -

92 II Limit MHSs
The structure of the ?>s-module Qx[l/(f(x) - t)] carried to QX[D] is of
the following form. Since
i\co
d,
v.co
.(/00-
¦V+l
¦(/(*)-.
(/oo-
¦ = (/00-(/00-t)
i\co
(/(*)-'>
,'¦+1
= /(*);
-— г-
the action of the ring Ds on Q>[?>] is given by the formulae
E.3.5) d,wiy.=(oDi+\
tcoEt = f(x)coLV - icoD'-1.
The differential d on the complex Qx[l/(f(x) - t)]
(i+l)\df/\w
(/(*) - O'+2
(d(/(x) - 0 = d/ - df, but d? = 0 in Qz/S) transforms to a differential d
on the complex QX[D] given by the formula
E.3.6) dicoD) = dcjLV - d/ Л wDM.
The pole order filtration P- on complex E.3.1) (see §4) transforms to a
filtration F' on Q>[?>] defined below in proposition E.3.11).
Taking into account the identification E.3.1) of the complex
DR-z/s(B[r]z), concentrated on the graph Г, with the complex QX[D] on
X, we obtain an expression of the Gauss-Manin differential system E.2.2)
in terms of the complex Q'X[D]
&x = nn
if
p*(DRz/s(Bir]Z))
E.3.7)
= W+l+kMQx[D]).
Let /: X -> S be the Milnor fibration of a singularity /: (C+1, 0)
* (C, 0). Then by proposition E.2.5) we have
E.3.8)
f &x=.
if
?n+\+k/
Consider the complex (f*Qx[D], d) = (K\ d). From formula E.3.6) it
follows that this complex can be considered as the single complex
associated with the double complex (K", du d2), where
E.3.9) Ю* = f^qIf, q^Q,Km= f*Qx[D],
5 The Gauss-Manin system of an isolated singularity 93
and the differentials are: di = d is exterior differentiation, d2 = -d/Л is a
Koszul differential, i.e. exterior multiplication by the form -d/
E.3.10)
о
T
T
T
T
T \
T-d/л
-o
T
*/*nj+1
N.
о- /.я
,0 A
T ^x
T
ч T-d/Л
f*ax Д/,?
T-dfA \d,
T - d/Л T- d/л
/,o;-' - ftax -ij
We summarize:
E.3.11) Proposition Let /: X -* S be the Milnor fibration of a singular-
singularity / Then the Gauss-Manin differential system J* & x =
¦%n+x+k(f*&x[D\) is the (n + 1 + fc)th cohomology of the single com-
complex (f*Qx[D], d) associated with the double complex E.3.10),
+?, d, -d/Л), and, in particular
E.3.12)
The complex /*Q^[?>] is a complex of ?>s-modules with operators of
multiplication by d, and t defined by formulae E.3.5). Under the identifi-
identification E.3.4) the pole order filtration P' corresponds to the Hodge
filtration F- on the complex f^Qx[D], the second filtration of the double
complex E.3.10), FT =®r*p+lKr's,
E.3.13) FPf*Q"x+l[D] =/*QJ+1 +/*QJ+1?» + ... + /*QJ+ID"'?.
This filtration induces a filtration F4 on Жх. Ш
The operator dt — D corresponds to a shift along the diagonal of the
double complex K-. It takes Fp to Fp~\

94
E.3.14)
II Limit MHSs
5.4 Connection between the differential system Жх and the Brieskorn
lattice Ж{0)
The Brieskorn lattice .JS?<°> = "Ж = f*Qnx+l /d/ Л dU^QJ) of an iso-
isolated singularity /is defined in A.5.1.3). It is a locally free ^-module of
rank ц extending Ж onto S. Obviously, FnK' = Kn+[fi coincides with
/*Q?+1 (F- is the second filtration of the double complex E.3.10)).
E.4.1) Lemma The inclusion
morphism
С К' onto F"K- gives an iso-
isoProof To prove this we have to find those elements со 6 /*&J+1 = FnK'
which are cohomologous to 0 in K\ a = d#. An easy search of the
diagram E.3.10) shows that these are exactly those со having the form
со = d/ Л d0, where в € f*Qx+1. ¦
E.4.2) Lemma The operator 9, is invertible on .Jf^-, 9,: Ж*- -* -^x-
The Hodge filtration on Жх coincides with the filtration by powers of the
operator d, applied to the lattice Жт:
F" Жх = d,F"JVx, • • • , F*-
x =
Proof This again is reduced to a diagram search in diagram C.10).
Let
E.4.3) Ж(к) = F*-kJffx = dkF".%fx-
Then the decreasing Hodge filtration on Жх, 0 С FnMx С
Fn~x3@x С ... , becomes the increasing filtration
E.4.4) Жт С Ж(Х) С ... i??(i) С i?f(*+1) С ... С Мх
E.4.5) д,\ Ж(к) ^ 3Z(M\
We have already met an extension of this filtration to the left in A.5.3.7),
D.6) Remark Scherk and Steenbrink [Sc-S] introduced the limit Hodge
filtration of the MHS on the vanishing cohomology based upon the theory
of As-modules with regular singularity at the point 0 6 S. The invertibility
6 Decomposition of a meromorphic connection
95
of the operator d, on .Жх.о then implies [Phi] that Жх — ®j=\.y^a>'4>,
а,ф-\,-2,
where.
= Ds/'Ds(td,-af.
6 Decomposition of a meromorphic connection into a direct sum
of the root subspaces of the operator td,. The F-ffltration and the
canonical lattice
6.1 'Block' decomposition
Let {.Ж, V) be a meromorphic connection on a disk ScC, regular
singular at the point 0 € S (see A.4.6.1), A.7.7.2)). .Ж is a sheaf on S such
that Ж\$ — Ж is a locally free sheaf of 6 S' -modules of rank [i, and the
zero germ Ж§ ~ .i?f is a vector space over the field Ж = C{/}[?~'], the
field of meromorphic functions with poles at 0, i.e. the field of fractions of
the local ring 6>sja (for simplicity we'll often write .Ж instead огуМ^). In
the following applications .Ш is the meromorphic Gauss-Manin connec-
connection and .Ж = Жл^ = Ж^ <g> г- 5,0Ж' is a localization of the Brieskorn
lattice Ж(й) wrt the maximal ideal »> С &$$¦ Let dt = Vd/d<: Ж —» Ж
denote the operator of covariant differentiation (the sheaf ^-M may be
considered as a sheaf of /^-modules). Let H = Ker V с Ж be the local
system of horizontal sections and Г be the monodromy transformation.
Recall A.7.7.2) that a connection V is regular, if there exists a lattice 3$
in ..Ж invariant wrt the operator td,. A residue of V wrt such a lattice is a
linear operator Res</ V on the vector space 5f/tJ? induced by the operator
tfv,. From the classical Sauvage theorem A.7.6.2) it follows from A.7.7.7)
that Ж decomposes into a direct sum Ж — ®\_хЖа"ч> of standard
blocks corresponding to the decomposition of the monodromy T into
Jordan blocks. Here r is the number of blocks and qt their sizes, Y^i=\4i
= fi = дхса.,хЖ- Or in other words, there exists a basis щ, ... , u^ in Ж
in which the operator tdt matrix consists of blocks each of the form
aE + N, where ? is a unity matrix, and N corresponds to a nilpotent
operator, Nit\ = иг, ...
diagonal),
F.1.1) 1дг(щ,
This also means that wrt the basis u\, ... , uq the connection V has a
matrix of connection coefficients of the form Г(/) = (\/t)(aE + N).
If y= (y\, ... , yq)' is a column of coordinate functions wrt the basis u,
then the horizontally condition for a section u-y reduces to a system
Nuq-\ = uq, Nuq = 0 (units are under the main
F.1.2)
у =—y, where R = -(aE + N).

96 II Limit MHSs
This system has a fundamental matrix of solutions
F.1.3) Y(t) = tR=eRl0Sl.
This implies that the monodromy on H_ is of the form
F.1.4) T = е2т1я = е-Мо-е-млг = TSTU,
F.1.5) Я = e~2ma is an eigenvalue of the monodromy T, and
F.1.6)
a = -T-r logA, N = -—- log Tu.
2m 2m
From the definition of a residue A.7.7.4) it also follows that wrt the basis
щ, ... ,uq or wrt the lattice ®f=1^s-«/, the residue is equal to
F.1.7) ResV = aE+N
and a is an eigenvalue of Res V wrt the lattice ®]=хб $щ.
6.2 Decomposition of a meromorphic connection Ж into a direct sum of
the root subspaces
First let Ж - .J6a'4 = .Жщ® ... ®.Жич, i.e. let Г have only one Jordan
block with the eigenvalue Я = e~2jna.
Let со be an element of Ж with coordinates q>(t) wrt a basis u,
со = u-q>(t), where cp(t) = (<p\(t), ..., cpq(t))' is a column of coordinates.
Decomposing cp(t) into the Laurent series cp(t) = ^cp^-t', where <pw are
constant columns, we obtain a decomposition
F.2.1)
со =
It appears that this decomposition has an invariant sense.
Recall that a root subspace La of a linear operator A on a vector space L
corresponding to an eigenvalue a is the subspace La = {со € L: 3 r,
(A - a)rco = 0}. Now note that on the subspace В = ©?=1Си,- С .Ж the
operator tdt = aE + N, and hence, the operator tdt — cl — N is nilpotent
and hence В С Са, where
F.2.2)
def
Cp "{со eJTy.3r, (tdt -p)rco = 0}
is the root subspace of the operator td, on ^ffl corresponding to an
eigenvalue /3. Moreover, the relation [dt, t] = 1 implies by A.7.7.6) the
following lemma:
F.2.3) Lemma On the space Д<*> = tkB = ®1=lC-tkUi we have
td, = (a + k)E + N.
6 Decomposition of a meromorphic connection 97
This means that 2?(i) С Са+к and, consequently, the decomposition
Ж = ®ktkB is the root decomposition of the operator td,,
F.2.4)
= ® Ca.
Here a runs over an arithmetical progression, a = — (I/2m log Я with step
1, which is infinite on both sides. Here the sum sign ® means that there
are inclusions ®aCa С .J6 с \[аС and Ж is generated by ®Ca over
?5 = ^5;c,o- Note that all spaces Ca have the same dimension dim Ca = q.
In general, when Ж — ®\=х.Жа''4' is a sum of blocks, we obtain:
F.2.5) Theorem If Ж is a meromorphic connection with a regular
singularity, then
where Ca is a root subspace of the operator td, on Ж, and a (for those
Са ф 0) runs over к arithmetical progressions a = —(l/2ni)Xj, where
X\, ... , X/c are all different eigenvalues of the monodromy T. ¦
F.2.6) Corollary Any element со €
form
is uniquely represented by the
where coa e Ca.
We call this the root decomposition of the element со, and the compo-
components coa are called the root (or homogeneous) components of order a. If
со = coa, the element со is called homogeneous.
Obviously, the operator of multiplication by t
F.2.7) /: Ca ^ Ca+i
is an isomorphism for all a. Since on Ca, td, = aE + N and d, =
(l/0(a? + N), we have that
F.2.8) d,: Ca - Ca-x
is an isomorphism for all а ф 0. In particular, on the subspace
F.2.9) SS= © Ca
a>-\
there exists an operator which is inverse to the operator dt,
F.2.10) д~1: Я -> %, 0,-0,-' = id.

98 II Limit MHSs
6.3 The order function a and the V'-filtration
Now let V be a quasiunipotent connection as is the case for the Gauss—
Manin connection, i.e. the monodromy operator Г is quasiunipotent: there
exist integers m and n such that (Tm - E)"+l = 0. Thus all eigenvalues
Я = e23ri/? of the operator Гаге roots of unity, j3sQ. Then all members of
the arithmetical progressions a = — (l/2OT)logA are rational numbers and
hence are naturally ordered.
F.3.1) Definition If со = ^ша € Ж, then a rational number
a((o) = min {a: coa ф 0}
is called the order of the element со. The first non-vanishing term
F.3.2) CO\ - COa(a>)
is called the main or leading part of an element со. The graduation
^S = ©Ca is called the root graduation.
F.3.3) Definition Natural decreasing nitrations, the V'-filtration and the
V>-filtration, are associated with the root graduation on Ж:
def
= {со e .Ж: a(co) 2= a} = © C&
and
def
V>aJ& = {со е Ж: а{со)>а) = © С/».
The corresponding joined objects are
F.3.4) Grav = Va/V>a~Ca.
From the description of ^S = ®Ca in F.2.5) it is obvious that all terms
of the filtration УаЖ and of У>аЖ are free ^-modules of rank /г
F.3.5) VaJ6 >
invariant wrt tdt, i.e. they are saturated lattices.
The lattice
F.3.6)
df
is called canonical.
F.3.7) In other words, S с ^Ж is a locally free ^-module extending the
sheaf 38 to the whole disk S, SS\S- = 3V. The sheaf ^ is uniquely
characterized by two conditions:
A) the connection V has a logarithmic pole on S?; and
6 Decomposition of a meromorphic connection
99
B) the eigenvalues of Res / V, i.e. the eigenvalues of the operator tdt on
5Slt3Z = Ф-KccsoCa, satisfy the condition -1 <a « 0.
The sheaf 5% is called the canonical extension of the sheaf ,Ж.
6.4 Identification of the zero fibre of the canonical extension 5? and the
canonical fibre of the fibration H
By a canonical fibre of the fibration H we mean a vector space H of its
multivalued sections over S', or a space of multivalued horizontal sections
of the sheaf 36 = Я®с &'s< = ^|ss or a space of single-valued horizon-
horizontal sections of the lift of 36 to the universal covering e: U —> S' B.4.8).
There is a natural isomorphism of the space H and the space
5% It2% = ©-KasoCa, the zero fibre of the canonical extension SS.
First let us consider the case of one block. Let the monodromy Ton the
space H have one Jordan block with eigenvalue X. Then for any choice of
one of the values of logarithm a = — (l/2;ri)logA we have an isomor-
isomorphism
H~Ca.
The mutually inverse isomorphisms cpa: Ca —> H and ipa: H —> Ca are
arranged in just such a way.
A choice of a corresponds to a choice of the basis щ, ..., uq in
in which td, = aE + N = -Ra- In this basis the space of
horizontal sections H is a space of solutions of the system of differential
equations y' = (Ra/t)y which has a fundamental system of solutions
F.1.3) r*¦, and an element A € H is represented by the form A = t^-yo,
where j0 6 C* is a constant column. Then:
F.4.1) q>a: Ca -» H takes a> = u-y0 ^ A = tKyu € H, A = Л".
Conversely, if Л € Я is a multivalued horizontal section, then the
multivalued family of matrices f1^ = ta-eNbgt having the monodromy
е-2тъ _ j-\ 'unravels'the multivalued section^:
F.4.2) s[A,a\ = r*°.A
is a single-valued section of the sheaf 36. An immediate calculation shows
(because d,A = 0) that
tdts[A, a] = as[A, a] + s[NA, a]
and hence, s[A, a] e Ca. Then:
F.4.3)
¦фа: Н —> Ca takes A >-* s[A, a].
The same formulae are valid when the monodromy T consists of some
Jordan blocks with the same eigenvalue X. We obtain:

100
II Limit MHSs
F.4.4) Let Ял С Я be a root subspace of the space of multivalued
horizontal sections corresponding to an eigenvalue X of the monodromy T,
dim Ял = fa. Then for any value of the logarithm a = -(l/2^i)logA all
root spaces Ca have the dimension dim Ca = fix and there are isomor-
isomorphisms
F.4.5) ipa: Hx^t Ca, A i-> y>a(A) = s[A, a] = tatNA,
where tN = cNhs'. A section s[A,a] of the sheaf Л? is called an
elementary section of order a corresponding to a multivalued section
A = A{t) e Ял.
Calculation F.4.2) shows that under the isomorphism ipa the operator
N = —(\/2л1) log Tu on Ял transforms to the operator tdt — a on Ca, ie.
there is a commutative diagram
td,-a
F.4.6)
(td, - a)-ipa = ipa-N.
F.4.7) In the general case let H = ©JL, Ялу be the root decomposition of
the monodromy T. Choosing for every Xj the value a.j = —A/2тп) log Xj in
the interval -1 < a} ^ 0 and applying the above construction to each
Ca,
block,
F.4.8)
which
we obtain
takes A =
an isomorphism
ip: H^* %/tJg -
E*=,4 e я = ©я,-
= ©
-1<а«0
to
к
F.4.9)
The isomorphism ip is extended to an isomorphism of sheaves
F.4.10) ip: Ss(H) = H <g> ES^ ST
and we obtain a natural trivialization of the sheaf %.
F.4.11) Remark The canonical extension % of the sheaf J& constructed
in B.6.4) is the canonical extension F.3.5) when all Xj = 1, and then
a = 0 (cf. formulae B.6.3) and F.4.5)). Recall that4 in §2 we made a
covering of the base S such that the monodromy became unipotent (all
Ay = 1).
6,5 The decomposition of sections a» € .Ж into a sum of elementary
sections
Let o) = ^2°>а € ^, a>a € Сa, be a root decomposition of the element со.
6 Decomposition of a meromorphic connection \ 01
By means of isomorphisms F.4.5) гра: Ял^ Са we can represent homo-
homogeneous components a>a in the form of elementary sections
F.5.1) coa = s[A%, a] = /-*¦•< = taeN 1о«' Аша,
where A® e Яя, А = e-2jria, are multivalued horizontal sections. In more
detail
F.5.2)
r2log2t
1! 2!
or using Varchenko's notation ([VI, AGV])
+ ... + N"-
F.5.3)
logt/
where Ашка = NkA%.
Now if у е Я* is a multivalued horizontal section of homological
fibration H*, the dual to Я, then
= (со, у) = Y.icoa, У) =
',e y)
where ala = (^, у).
We thus obtain the decomposition into series of period integrals
Iy,co(t) = Jy(t)CO (see A.9.4.4)), from which one usually begins the exposi-
exposition. The advantage of the above approach (the inverse order of exposition)
lies in its invariant character.
From corollaries of the Malgrange theorem (A.9.4.5) and A.9.4.6)) we
obtain
F.5.5) Corollary If we :^-x\ then a(co) 3= 0. If со е .^@), then
a(co)>-l.
Thus,
F.5.6)
6.6 Transfer of automorphisms from the Milnor lattice Hto the
meromorphic connection ..Ж
Let Я be the cohomological fibration of a singularity /: (C+1, 0)
-> (C, 0). Then Я = Яя(Дг0О, С) is the cohomology of the canonical
Milnor fibre B.4.6). In the study of singularities which follows, the
meromorphic connection .Ж, a topological invariant of a singularity, will

102
II Limit MHSs
play greater role. Ж is a 'universum' in which all Brieskorn lattices
of singularities /of a given topological type live.
F.6.1) The isomorphism F.4.8)
V>: Я те © Сц С ^ =
-1<а*0 а
enables us to carry different structures from Я to Ж.
F.6.2) We will now carry automorphism of Я to automorphism of Ж.
Obviously, any automorphism g € GL(H) is uniquely extended to an
automorphism of Ж as a (Ж = C{t}[r]])-modnle in such a way that the
diagram
Я ^ © Са С ->Ж
Я
CaC
is commutative. We denote by ip(g) 6 Aut.^ this automorphism of Ж
which on ®_i <«so Ca is equal to rp{g) — ip-g-ip~l.
F.6.3) Lemma The automorphism on Ж corresponding to the mono-
dromy Гоп Я is
Proof It follows from F.4.6) that the operator td, —aonCa corresponds
to the operator N = -(l/2m)logTa on Я A power of td, -a corre-
corresponds to a power of the operator N. Hence, the operator on Ca corre-
corresponding to the monodromy operator T and coinciding with
XTU = X s-bAN on Яя is given by
^ -2лЦ1д,-а) _ ^ e2ma e-2mtd, __ ^-Imtd,
It follows also from F.4.6) that the operator e'2™'9' defines ip(T) in the
sense F.6.2) on Ж. ¦
Ж is in a natural way also a C[3f]-module.
F.6.4) Lemma An automorphism g € GL(H) commutes with T if and
only if V(g) commutes with d,, i.e. V(g) is an automorphism of ^-Ж as a
C[3,]-module.
Proo/ If V(g) commutes with dt, then V(#) also commutes with td,
7 The limit Hodge filtration 103
because ip(g) commutes with t. Then V(g) commutes with e~2jn'a' =
V(r), and, consequently, g commutes with T. Conversely, if g commutes
with T, then ip(g) commutes with е~2л'9', and, consequently, with td,.
However, since ip{g), as an automorphism of the Ж -module, commutes
with t, ip(g) commutes also with d,. ¦
7 The limit Hodge filtration according to Varchenko and to Scherk-
Steenbrink
At first the MHS on the vanishing cohomology of an isolated singularity
/: (C+1, 0) -> (C, 0), i.e. on the cohomology H"(XX, C) of the (canoni-
(canonical) Milnor fibre, was introduced by Steenbrink C.4.4) by means of an
embedding of the Milnor fibration X —» S in a projective fibration Y —> S.
By a result of Scherk C.5.1), this embedding can be chosen such that the
homomorphism Hn(Yoo, C) —> Нп{Хж, С) is an epimorphism and the
limit MHS on Я»AЮ, С) is a quotient of the limit MHS on H^Y^, C).
The weight filtration W. on Нп{Хж, С) is a quotient weight filtration on
Я"(Уоо, С) and is defined by means of the nilpotent operator
N = — (l/2;ri)log Ги, the logarithm of the unipotent part of the mono-
monodromy on Н"(ХЖ, С) C.5.2). It is explained in C.5.4) why this filtration
has different 'centres' on subspaces H"(Xoo, C)^i and H"(Xoo,C)x=\.
Varchenko proposed the idea of introducing the Hodge filtration F~ on
Я'Ч-Л'оо, С) directly (by means of integral asymptotics), without using an
embedding in a projective family. Following this idea, Scherk and Steen-
Steenbrink [Sc-S] introduced the filtration F- in a different way. We begin by
introducing the filtration F~ = Fss on H"(XX, C) according to Scherk-
Steenbrink because it gives a genetically more naturally motivated con-
construction of the limit Hodge filtration.
7.1 Motivation of Scherk-Steenbrink's construction of the Hodge
filtration
The sequence of steps leading to the construction of filtration F'ss is as
follows.
A) The sheaf Ж = H®S' (9? = Ж^Х'/S') has a canonical exten-
extension 5? on S, and there is a canonical isomorphism F.4.8)
V: Я = Н'ЧХ^, C)^ %lt% = Ф Са.
=t<x

104
II Limit MHSs
of the space H"(XX, C) considered as a space of multivalued horizontal
sections of the fibration Я and the zero fibre of the sheaf 2".
B) The Hodge filtration F' on H"(YX, C) according to Schmid (see
§2.2) (coinciding with the Hodge filtration according to Steenbrink) is the
limit Hodge filtration on the cohomology of the projective varieties
H"(Y,, C) B.6.6). Let a bar denote an object for Y -> S analogous to the
corresponding object for X -> S, ~Ж = Я®с5. Г' s = J%qR(Y'/S') and so
on. Then the limit Hodge filtration T- on Я = H"(YX, C) is a filtration
induced by the filtration on the zero fibre c/C/tc/ of the canonical
extension of the sheaf .Ж on S, though after a preliminary lift of the base
S —» S 'killing' the semisimple part of monodromy Ts. The filtration on if
is obtained by an extension of the Hodge filtration .T- on Ж, where .9~- is
the variation of the Hodge filtration on H"{Yt, C). The consequences of
the necessity of a base lift 5 —> S are discussed below in E).
C) One can calculate the cohomology of a hypersurface by means of a
complex of principal parts of differential forms on the ambient manifold
with poles along this hypersurface D.3.5). The sheaf.Ж is identified with
the sheaf Jgy]s,, Л'у = Un+^(DR^/S(B[T]1)), where fcZ=7x5is
the graph of the morphism 7-»5 E.1.4). The Hodge filtration is identified
with the pole order filtration. We obtain an extension of the Hodge filtration
.9~' on Ж to a filtration on the sheaf .Ж у and in particular on the canonical
extension Я> = ¦& of the sheaf. Ж to S.
D) We can repeat the construction in C) for the morphism f:X—>S
(§5). We can use this to introduce the limit Hodge filtration on the zero
fibre of the canonical extension % of the sheaf 3% from 5" to S, without
using an embedding into the projective family. Thus, we have to consider
the differential system E.3.12)
where
x x
, coff
l\U)
-O'+1J'
to take its Hodge filtration F~ E.3.13) induced by the pole order filtration,
and to consider the induced filtration on the canonical extension if с Ж х
of the sheaf.Ж on 5".
We have an inclusion of the Brieskorn lattice .Ж@) С .Жx onto the и-th
term of the filtration F"MX, ш >-* [co/(f(x) - t)] E.4.1). Moreover, the
remaining terms of the Hodge filtration are obtained from .i^<0) = Р".ЖХ
by applying the operator д„ Fn~k.Mx = dktFn.Xx = д)Ж^ E.4.3).
7 The limit Hodge filtration
105
Thus, the Hodge filtration is defined by the inclusion of the Brieskorn
lattice .Ж@) С Жх.
Consider a filtration on % С -Жх induced by F' on Жх,
fpz = fp.mx n jgf = д"~ржт пя = д"-"^ n-p
The last equality is valid because % = V>~1 and the operator dt takes Va
to Va~x. By definition FPSZ is generated by elements of the form
d"~ps[ca], where s[co] = Yla»a(aj)>-is[c4, a] € J??@) are elements of or-
order a{wi)> n — p - 1. On SC/t^ = Ф-1<а«оСа the induced filtration is
of the form
This is the desired limit Hodge filtration for the unipotent monodromy
T = Tu. Scherk and Steenbrink defined it in this way in the initial version
of the paper [Sc-S]. However, in general, it is still not the desired filtration
[Sa2, Ph2] because we have not yet dealt with the base lift 5 —» S.
E) The necessity of having a base lift S —» S 'kill' Ts leads to the
following. In the above definition instead of using the elements d"~ps[co]
to generate FPM we have to take their leading parts d"~ps[a), а(со)]. Let
us explain this.
Without the base lift the filtration term FPSA is generated by elements of
the form Q) = 1?a>-\tatNA" such that dj(n~p)<o € .Жт, and for the
filtration induced on Я = Фа Ял ^ ®-i<a=soCa the subspace FPH is
generated by the elements J^-i < aso^a •
According to Schmid's theorem we have to do the following. We have to
take a covering ж: S —> S, t = ~tm 'killing' the semisimple part of the
monodromy, T™ = id, and making it unipotent, t = Tm = T™ = fa (a
tilde denotes the corresponding object on S). We have to take the inverse
image .T' on :Ж = л*.Ж of the Hodge filtration .9"- on 3% and to
consider a filtration on ^ induced by the filtration on S?,
where n*{Fp,SZ) = (Fp5f)®rs &s- Then, finally, we have to take the
filtration F' induced on the zero fibre % /t%' ~ Я. This is the limit Hodge
filtration according to Schmid. Since the monodromy f is unipotent, we
have that in the decomposition ,^~фа>_1Са all a are integers,
X = ®keN0Cic, and the mapping i? —> s?/t% reduces to a> =
Хд»ош* ь^ cbQ. The filtration n*(FpS5} = (Fp%)®r-s 6-s is generated
by elements from С{7}лг*ш = С{1}-^а>_11та1тМА%. То obtain
Fpk = n*(Fp,%) П SjC we have to take elements of degree > 0 in ~t. We

106
II Limit MHSs
see that the subspace Fp of 56 ftS% ~ // is generated by elements A^^,
i.e. by first summands in ^2-\<a^oA".
F) Lastly the final note: instead of operating in the context of the
Gauss—Manin differential system Jlfx we таУ operate in the more usual
classical context of the meromorphic connection Ж — ,Ж@' <8>^s Ж. This
follows from the fact that the canonical lattice S? lies in .%§x as well as in
Ж, if С .Жх and 5? С Ж. We'll talk about the relation between Mx
and Ж in more detail in the supplement to this section in G.7.9)—G.7.11).
The sheaves SZ', 36x and Ж are topological invariants of a singularity
defined by the Jordan structure of T on H. Analytic invariants of a
singularity /(moduli) are reflected in the inclusion of the Brieskorn lattice
to the canonical lattice, ^@) С 56, which defines among other things the
MHS on the vanishing cohomology.
The above explains the following definition:
7.2 The definition of the limit Hodge filtration F$s according to Scherk—
Steenbrink
Let Ж = ®Ca be the meromorphic connection of a singularity /
if = V>""' = фа> -i Сa С Ж be the canonical lattice, and J^@) be the
Brieskorn lattice. By F.5.6) we have J%<-0) С 5§.
G.2.1) Definition The Hodge filtration F' = F'ss according to Scherk—
Steenbrink is defined on the zero fibre 56/t56 — ®-i<asoCa on each Ca
separately,
FpCa,
G.2.2)
df -
where
G.2.3)
G.2.4)
¦ppr — r)"'
-¦ VPn.%fi0)/V>P С G/уЖ - Cp is a subspace gen-
generated by the leading parts of geometric sections s[co] € 36^ of order /?.
Since i^@) cV>~\ws have Gr^.J^@) = 0 for 0 « -1 and hence:
G.2.5) FpCa = 0 for p > n, FpEf/t56) = 0 for p > n.
If /? > — 1, then there is an isomorphism dj1: Cp^* Cp+i, inducing inclu-
inclusions
7 The limit Hodge filtration
(recall that d~x takes .Ж@) into itself and dj\.MiQ)) = J
A.5.3.4)).
107
by
G.2.6) We can schematically visualize the inclusion .Ж^ с Ж, more
exactly Gry.y^@) С вгЖ, and the Hodge filtration F- on Ca,
—l<a«0, by means of the following diagram [Hel] (see picture
below). Let us picture S? = ©a>_i Ca as an infinite set of columns C^ of
height
G.2.7)
dim Cp = dim Hx = Их, where Я =
J
F"
T-
s
s
37'
-—-
"«+2
s
в ^
37'
"a
„
¦ч
T
u
с с
37'
сГ
-
и
37'
*
37'
1 a + 2 з ¦» n-p
•* о
j. +
I
The columns Cp and C^+i are identified by means of the isomorphism
d~x. The parts of the columns corresponding to Gr^y.M^ с Cp are
hatched and the parts corresponding to dy1Grey~l.^fm =
GrBv.^-1) С G^^) are shaded. The 'figure' Gr-у.Ж^^ is obtained
from Grv(:Mm) by a shift of 1 to the right. Part of G>>(J^@)) correspond-
corresponding to one eigenvalue Я of the monodromy is shown in the picture above,
and as an example the whole of G/-K(.^@)) is shown for the singularity
Ej — Xg = Гг,4,4 in picture on the next page.

11Limit MHSs
c_% c.Vl c_Vi
-1 -'A -V, -V, 0
7.3 The Scherk-Steenbrink theorem
G.3.1) The filtration Fss on Grv{Z/t?) — ®_i<a<oCa is carried
to H"(XX, C) by means of the isomorphism V: #"(^oo> С)те
Jgyfif ~ Gr-Y{J?11&\ By construction, t/>: Н"(ХЖ, С)а -^ Ca> where
Я = e~2jria, -1<а«0, Hn{Xx, C)x is a root subspace of the operator Г,
corresponding to an eigenvalue A. The filtration F- being defined on each
Ca separately, F" is invariant wrt Ts-
G.3.2) There is a weight filtration W = W(N), N = -(I/2m) log Tu, on
Н'ЧХоо, С) defined in C.5.2). Under the isomorphism ip it corresponds to
the filtration W = W{N) of a nilpotent operator N on the space
Ф-i<a«oCa, where N = td, — a on Ca. The filtration Wis also invariant
wrt Ts by construction.
Let Fj denote the Hodge filtration on Hn(Xoo, C) according to Steen-
brink in theorem C.4.4), which also coincides with the limit Hodge
filtration according to Schmid ([S3]).
G.3.3) Theorem ([Sc-S]) The Hodge filtration Fss according to Scherk-
Steenbrink on Н'ЧХоо, С) coincides with the Hodge filtration Fs accord-
according to Steenbrink. Consequently C.4.4), the filtrations W. and F- = Fss
define an MHS on Н"(ХЖ, С). This MHS is invariant wrt the semisimple
part of monodromy Ts, and it decomposes into a direct sum of MHS
Let us consider examples of calculation of the MHS on the vanishing
cohomology.
G.3.4) The MHS in the case of quasi homogeneous singularities
Let f(x) € С[ло, ..., xn] be a quasihomogeneous singularity of degree 1
with weights wtjc/ = wt. We use notations from A.5.5.1-1.5.5.3). The
7 The limit Hodge filtration
109
differential forms com = xm dx, m € A, \A\ = /л, represent a basis of the
<^5-module Ж@) = ®meA^ sc>m- We use the same symbol to denote
differential forms, geometric sections defined by them, elements of the
space i?/ti6 = ®_i<a*o = C, etc. In A.5.5.4) we showed that
wm € JffW are eigenvectors of the operator tdt,
dm € Ca(m)-\-
Thus the nilpotent operator Af is zero and the weight filtration is as follows:
о с ж„_, с wn с wn+l = c,wn= e ca, wn+l/wn = Co.
-Ka<0
The Hodge filtration F' = Fss, С = F° D F1 D ... D F" D Fn+l = 0,
is defined as follows. Let us take the partition of the set of indices (= the
set of exponents of monomials) A into subsets, A = L)AP, where
Ap — {m & A, n — p — \< a(m) — 1 =? n — p). We have that if m € Ap,
then rjm = d"~pct)m € Ca, where — 1 < a =? 0. The elements rjm generate a
basis of the lattice 5f, Ж = ®тел^ sVm, and correspondingly of the space
С = ©телС»7т. Then by definition
FPC= © Cr)m, where A*p = ApUAp+l U... U А„.
€A
G.3.5) The MHS in the case of singularities T' M<r ([Sc-S]) The unimodal
singularities of series TM>r are defined by equations
fix, y,z) = xp + y4 + zr + axyz, а ф 0,
where p~x + q~x + r~x < 1 ([AGV]). These are simplest not (semi)quasi-
homogeneous singularities. For them /x = p + q+r— 1, and the mono-
monomials Mi
1, xyz, xki0<k<p), yki0<k<q), zki0<k<r),
form a basis of the Jacobian algebra Qf. The differential forms w, = M,co,
1 =? / «fi, where ш = dx Л dj Л dz, form a basis of the vector space
Qf = .y%W/.%f(-l\ Applying the method developed by Scherk [Scl], one
finds that a basis of the canonical lattice J? consists of the forms
о, td,eo, d,ixkeo) @ < к < p), d,iykeo) @ < к < q),
d,izkeo) i0<k<r),
(by (III.3.4.14) we can replace td,ct) by d,ixyza))). Moreover, except for a
few small values of p, q and r, the operator tdt on С = S§/tSZ has the
following form relative to this basis:
td,itd,co) = 0; td,id,xkeo) = ^—^-d,xkm forO<k<p;

по
II Limit MESs
tdt{dtykw) =
Э,уксо forO<k<q;
к — r
td,{d,zkco) = d,zkco for 0 < к < r.
Thus, this basis is a Jordan basis for the operator tdt. The space
С = ®-]<a^oCa = C-to © Co, where the space Co is generated by vectors
со and td,(o. The weight filtration Won С is the weight filtration W(N) of
the nilpotent operator N with center n = 2 on C^o and with center и = 3
on Co- On C^o the operator N = 0 and hence ff i C^o = 0, W2C-to = C^o.
On Co the operator N Ф 0 and N2 = 0, and we have W\ Co = 0,
Ff2C0 =W3C0 = Ctd,co, W4C0 = Co. Therefore the weight filtration is as
follows: 0 = Wx С W2 = Wi С W4 = C, where ff2 = C^o Ф Ctd,co is
generated by all base vectors different from a>. By definition of the Hodge
filtration we have С = Fl D F2 Э F3 =0, where F2 is the subspace
generated by w.
7.4 Varchenko's theorem about the operator of multiplication by fin
By the definition of the filtration F- in G.2.1) we have
G.4.1) GrpCa = FpCa/Fp+lCa
where Qf = 3&®/j&-V = QJ+J/df Л Qnxfi, and V~ also denotes the
filtration induced by Vй on Qf. Thus,
G.4.2) d"~p-- Gr^Qf^* GrpFCa, where /3 = a + n - p,
and
G.4.3) ®d"~p: Gr-yQf=*GrF( ® Ca) = GrFHn(X0O).
a,p \l<a«0 /
Moreover, since the operator N is equal to td, — a on Ca, the mapping
induced by it, GrFN: GrFCa -> Gr^~'Ca, coincides with td,, and we
have a commutative diagram
G/K+IQ
1
G.4.4)
7 The limit Hodge filtration
111
However, under the identification of the C{f}-module Q/ with QjV/
d/ Л Q."x 0 the operator of multiplication by f reduces to the operator of
multiplication by /(*). This implies the following theorem [V4, Sc-S]:
G.4.5) Varchenko s theorem The operator of multiplication by f(x) on
GryQ.f and the operator N =-(l/2m)logTu on Н"(ХЖ) ~
ф_1 <a=soCa have the same Jordan normal form.
7.5 The definition of the limit Hodge filtration F' on H"(X <*,) according
to Varchenko
The filtration F'ss on H"(XX, C)-^®-x<a^0Ca was defined by Scherk
and Steenbnnk after Varchenko had defined the asymptotic Hodge filtra-
filtration Fya. Varchenko defines this filtration F^ on fibres of the cohomo-
logical fibration H, — H"(Xt, C) or the corresponding filtration .9~'Vz on
the sheaf Ж = H <g> 6'$' in such a way that ,^f^a is generated by leading
parts co\ = s[co, a(w)] of geometric sections s[co] € .^@) of order
a{co) ^ n — p. In terms of the canonical extension ie = V>~] of the
sheaf Ж this means that on % the induced filtration .^"уа is generated by
sections tkco\, where к is such that -1 < a{tkco{) =s 0. In other words:
G.5.1) Definition The Hodge filtration F' = F'Va according to Varchenko
is defined on the zero fibre S?/t.% — ®-\<a^oCa and on each Ca
separately
G.5.2) F
and
G.5.3)
pP q _ f-(n-p)Qra+n-p
7.6 Comparison of the filiations Fss and FVi
From formulae G.2.3) and G.5.3) it follows that to get filtrations F'ss and
irVa on 561 tSZ we have to take the same geometric sections
s[oj] € :M(K), n- p-\< а(а) s? n - p,
take their leading parts
s[co, ft] = ^1МАШ, a(co) =p = a + n-p, -Ka =s 0,
and then carry them into .5?/tJ? = Ф-\<а^оСа firstly by d"~p and
secondly by dividing by t"~p. It follows from F.4.2) that dt: Cp -» C^_i
takes s[A, ft] to

112 П Limit MESs
G.6.1) d,s[A, p] = Ps[A, Р-Ц + s[NA, /5-1]
= s[(fiE + N)A, P - 1].
Hence,
G.6.2) drPs[A, p] = s[((a + \)E + N)
X ((a +2)E +N)...(J3E + N)A, a]
= ((a + \)E + N) ... фЕ + N)A-r(n-p)s[A, p]
and, consequently,
G.6.3) F&Ca = ((« +l)E + N)...(J3E+ N)F$tCa.
This formula shows that
G.6.4) FyZ/tSS) = F$AZ/t&) mod N,
or
F^tf"^) = FPaHn(Xoo)modN-H"(XO0).
Since N(Wk) С ^t-2. we obtain:
G.6.5) Corollary The filiations Ffs and F^a on Grf H"(Xao) =
Wk/Wk-\ coincide.
This corollary and theorem G.3.3) imply Varchenko's theorem [V3]:
G.6.6) Theorem The filtrations W. = W{N) from C.5.2) and F' = Fva
define an MHS on the vanishing cohomology Hn(Xoo, C)
We emphasize once more that the idea of defining the Hodge filtration
F- = Fya by means of the Brieskorn lattice Ж<0) (and the proof of this
theorem) are due to A. N. Varchenko.
7.7 Supplement on the connection between the Gauss-Manin
differential system Ж х and its meromorphic connection ^M
In this review we have tried to avoid appealing to the theory of ?>-modules.
Nevertheless we want to give some feeling of how the sheaves Ж х and
Ж are related. Let S С С be a disk centered at the point t = 0, D =
Ду0 = C{t}[d,] be the ring of differential operators in one variable, and
Ds be the sheaf of differential operators on S. If M is a ?>-module then an
operation of localization M(t) = M®r Ж = М[Г{] is defined. Here
Ж = C{t}[t~l] is the field of meromorphic functions - the field of
fractions of the ring &' = &s,o = C{?}. M(t) is naturally a ?>-module,
where the action of дt on M(t) is defined by the rule of differentiation of
7 The limit Hodge filtration 113
fractions. One can find in [Sa7, Sa8] a survey of the theory of ZMnodules
finite over Д with its localization M(l) being a regular meromorphic
connection in the sense A.7.7.2) (i.e. the theory of regular holonomic D-
modules). Let MTh(D) be the category of such ?>-modules. ?>-modules
Жх = M and .Л = M(f) e Mrh(?>) (see E.4.5)), and by E.4.2) the
operator of multiplying by d, is invertible on Жх, and the operator of
multiplying by t is invertible on .//6. The main fact is that any such D-
module is a direct sum of indecomposable ?>-modules
G.7.1) Жал = D/D(td, - aL = D/D[d,t - (a + 1)]*.
One can assume that up to isomorphism (see the lemma about shift in
A.7.7.6)): if a ? Z, then M = .Ла-\ a € A', where Л' с С, a set of
representatives of non-zero classes of C/Z; and if a € Z, then either a = 0
(for a s= 0) or a = -1 (for a < 0), i.e. we have:
G.7.2)
either M = .Ж0'4 = D/D-(tdt)q, or M = J/r1'4 = D/D{d,tL.
Moreover, for a ? A' both operators t and dt are invertible on the D-
module .Жа'4, д, is (and t isn't) invertible on Ж%ч, and t is (and d, isn't)
invertible on ,Жх>ч.
To understand better what happens when we pass from Жx to Ж, i.e.
from M to its localization M(/), we consider a ?>-module M = ^#°'« and
for simplicity we'll assume that q = 1. In general there is a homomorphism
of a Z)-module Mto its localization M(/) and we have an exact sequence
G.7.3) 0 -> XM -»¦ M Л M(o -> M* -> 0,
where rM = Ker q> is an 6> s,o-torsion of M, and M* = Coker <p.
The following simple lemma plays an important role in clarifying of the
structure of ?>-modules.
G.7.4) Lemma If Q = qro(O + ?i@^ + • • • + qm{i)d? 6 ?> is a differ-
differential operator of order m, then for any a € С it can be presented in the
following form
Q = dmd? + dm^d?~l +... + did, + c(t) + P(tdt - a),
where dk e С are constants, c(t) — Y^%ocjtJ e ^s,o is a holomorphic
function, and P e Da differential operator.
Proof This easily follows from the commutator rule
G.7.5) [dt, t] = 1, i.e. d,t -td,= \.

114
II Limit MHSs
Thus, let M = ЖаЛ — D/D(td, — a) — D-u, where и is a generator
satisfying the relation (td, — a)u = 0. Let us consider a D-module Ж-е,
which is a vector space of dimension q — 1 over the field Ж = C{ ?}[?"']
with a generator e, on which d, acts such that (td, — a)e = 0, i.e.
G.7.6) d,e=% and then <?> a<*»-0 •••<(«-'" + l)e
(i.e. we can assume that e = ta and 9, is the differentiation in t). Then it is
easy to see that Ж'-е = M(,j is the localization and the homomorphism
<p: M -> M(t) = Ж-е, u^e, takes Qu = (dmd? + ... + dld,+
c{t))umod D{td, - a) to the Laurent series
-+c(t))e.
G.7.7)
From this formula we obtain:
G.7.8) Corollaries
A) If a fi Z, then q> is an isomorphism, i.e. M coincides with its
localization M(,j and, moreover, d, is an isomorphism, d,: Cp-^i Cp~\,
where M(/) = ©C^, Cp = C-tke is a root subspace of the operator td, on
M with the eigenvalue /3 = a + k.
B) If a = -1 (a e —N), M = ^~1>1, then again <p is an isomorphism,
but d, is not invertible because Cp -& Cp-\ is an isomorphism for ft ф О,
and 9»: Co —» C_i is a zero map.
C) If a = 0 (a € No), M = ^C0>1, then q> is not an isomorphism:
Kercp = {(fl/md,m + ... + dxd,)u}, and lm<p = {c(f)e} = C{r}eC .Же
and Coker <p = C{t}[t~l]/<C{t} = M* is a space of principle parts of
meromorphic functions.
G.7.9) Let us consider this third case M = ,/Ж0'1 — D/Dtd, in more
detail. By G.7.4) elements Qu e M can be written in a form
(dmd™ + ... + d\d, + c(t))u. Such a representation corresponds to a root
decomposition of the operator td, on M:
M = ®Ma = ... © M-i © Mo © Mi ... Э ... + d\d, + c0 + ci / + ...
d, is an isomorphism: the central place is
d,\ Мо^з M\, c0 y-> cod,.
Conversely, the operator t is not an isomorphism: since td, = 0, we have
that t: M_i —v Mq is a zero map. Consequently, ©ae_^Ma = Kercp is а
C{ ^}-torsion.
Let M(,) = ®Ca Э J2cktk, к s= A:q, be a root decomposition of M(,).
5 Spectrum of a hypersurface singularity 115
We see that in going from M = ®M* to M@ = ©C^ in G.7.3) we 'cut
off the tail' ф*<оМ* = Mr in M and 'change' it by ф*<оС* = M*.
lies
()
Note that the canonical lattice <? =¦ V>~x =
in Mas well as in M^,y.
G.7.10) iT с M and J^f С Mw.
G.7.11) Therefore, if for a ?>-module M = ©.,Ma-q the operator d, is
invertible, then the transition from M to M(,j consists in changing the
summands ^Q'q by .Ж~х'4. (Conversely, if the operator t is invertible,
M = M(,), then the transition to a D-module with invertible operator d,
(microlocalization) consists in exchanging .Ж~х'4 and л
8 Spectrum of a hypersurface singularity
The spectrum Sp(f) of an isolated singularity /: (Cn+1, 0) -»(C, 0) is a
set of ^ rational numbers a.\, ..., a^, which is its most important discrete
invariant. It codes the relation between the semisimple part of the mono-
dromy land the asymptotic Hodge filtration F- of the MHS on H^X^).
The numbers ay- = — (l/2m) log A/ are the logarithms of the eigenvalues of
monodromy Ay = e~2maJ, and the choice of values (branches) of logarithm
is defined by the filtration F~. More detailed information about the MHS
on H"(Xoo) is given by a set Sppif) of spectral pairs (a,, lj), which, in
addition to the Hodge filtration, takes into account the weight filtration W.
on Hn(XO0). Giving the Sppif) is equivalent to giving the Hodge numbers
hpx'4 of the MHS, which are the main set of discrete invariants of a
singularity. The initial definition of the spectrum for isolated singularities
[S3] was extended by Steenbrink to the case of non-isolated singularities
[S6].
8.1 The definition of the spectrum of an isolated singularity
Let F-, H = F° D ... D Fp 3 Fp+x D ..., be the Hodge filtration on
the vanishing cohomology H = H"(X0C) of an isolated singularity / The
filtration F' is invariant wrt the action of the semisimple part of mono-
monodromy rs G.3.3). Hence, Ts acts on GrpFH = Fp/FP+x and GrpFH =
®k{GrpF)x, where {GrpF)x = {GrpFH)x — GrFHx is the eigensubspace cor-
corresponding to X. Let
(8.1.1) цр

116
II Limit MHSs
x. ~ i"a
Then Y^pt*p — t1 is tne Milnor number, ]Гд,"Г = Ир> and
the multiplicity of an eigenvalue Я, ,мд = dim Щ. We can consider the set
of eigenvalues of a monodromy Aj, ..., A^ as a 'divisor', i.e. as an element
of a free abelian group Z(C) with generators (A), A S C,
(8.1.2) (A,) + ... + (Л„) = ][>-(A) = ?>f-(Л).
A X,p
Thus, all eigenvalues A are distributed to levels p according to the filtration
F-.
Now to each eigenvalue A we juxtapose its logarithm a —
-A /2m) log A. Since A is a root of unity, a is a rational number defined
modulo an integer. We normalize a (choose a value of the logarithm)
according to the level p of A wrt F~ by the condition
(8.1.3)
a = г log A, n — p — 1 < a < и — р
2л1
-1 0
n — p - I n — p
n - 1
We obtain an element of the group Z(Q)
(8.1.4)
which is called the spectrum of the singularity /. The numbers aj are
called spectral numbers, and na = fi^ the spectral multiplicities.
By G.4.2) we also have
(8.1.4') tip = dim GrpFCa = dim
where ft — a + n - p, — 1 < а *? 0, and
(8.1.4") dim G^ i^°> = na + na+1
(see picture, p. 107).
= f{
np = f
(8.1.5) The choice of logarithm values (normalization) (8.1.3) is motivated
by the definition of the filtration F~ in the previous section, G.2.3),: Fp is
generated by leading parts of geometric sections of differential forms
weJ^@) with orders a(co) ? (n -p - 1, n - p]. Thus, if cou ...,
(OftP € ~^@) are such that n — p—\<a(<wy) < n — p and s[(Oj, a(ct)j)]
form a basis of Fp/Fp+X, then the orders a(a>i), ..., а(шир) are the part
of the spectrum belonging to (и — p - 1, n - p].
8 Spectrum of a hypersurface singularity 117
The spectrum defined as a set of (i rational numbers, or as a 'divisor'
Sp(f) = Xlna(a) ? ^(Q) can als0 be considered, following M. Saito
[Sa9], as a fractional Laurent polynomial
(8.1.6) Sp(f) =ta> + ...+ ta> = Y^nata & Z[tl/m, Г1/т],
a
where m is a common denominator of the numbers а,, Г™ = id.
(8.1.7) Remark M. Saito normalized logarithms in a different way: the
spectral number a corresponding to an eigenvalue A of level p satisfies the
condition
(8.1.8) a = ——; logA, n - p<a *? n -/? + 1.
2от
If we denote by Sp(f) the spectrum of a singularity according to Saito,
then Sp is obtained from Sp by 'a shift by 1 to the right' or
(8.1.9)
The spectral numbers contained in Sp(f), i.e. numbers a + 1, Saito calls
the exponents off.
8.2 The spectral pairs Spp(f)
The spectral pairs, or characteristic pairs as Steenbrink called them in [S3],
give more detailed information than the spectrum does. They take into
account a distribution of eigenvalues of monodromy A not only wrt the
Hodge filtration F\ but also wrt the weight filtration W. Since both
nitrations F' and W. on the space H =¦ Hn{Xoa, C) are invariant wrt the
semisimple part of monodromy Ts, Ts acts on Gr^Gr^^H and
GrpFGrwp+qH = @iGrpFGrwpJrqHx. In C.5.5) we denoted Hodge numbers
by h{*= dim GrpFGrwp+qHx.
(8.2.1) Definition Let us juxtapose these hf'4 eigenvalues A and the set of
«a,i = hp'q spectral pairs (a, /), where a = -(l/2m)logA is a spectral
number normalized by the level p by formula n — p — Ка^л- р
(8.1.3), and / is a weight number
(8.2-2) l={P + «' %X/\>
K ' \p + q-l, if Я = 1.
As in the case of the spectrum we can consider the set of spectral pairs
Spp(f) as an element of an abelian group with generators (a, 1) € Q X Z,

II Limit MHSs
118
(8.2.3)
Obviously, giving Spp(f) is equivalent to giving the set of all Hodge
numbers hp'4', and their spectral multiplicities are
(8.2.4)
«a =
, na,l-
8.3 Properties of the spectrum
The symmetry of Hodge numbers in §3 implies the following symmetries
of multiplicities of spectral pairs of an isolated singularity /:
(C+1, 0) -> (C, 0).
(8.3.1) Proposition
@ "a,l = »2n-\-l-a,l,
(Ю naj = na-n+it2n-i,
(iii) «o>/ = nn-i-a,2n-t-
Proof The proof of the first formula follows immediately from the
symmetry C.5.6) h%'q — hjP, and that of the second from C.5.7)
*?* = h"~x4'"-p and C.5.8) hp'q = h1+x-4'n+l-P. The second number of a
spectral pair is defined by formula (8.2.2) in such a way that for both types
of second symmetry of hp'4 the symmetry for spectral pairs is given by the
same formula. The third formula in (8.3.1) follows from the two first.
Similarly, any two of these formulae imply the third. ¦
(8.3.2) Corollary (symmetry of the spectrum) The spectrum Sp(f) of an
isolated singularity /: (C+1, 0) —> (C, 0) is symmetric wrt the point
(8.3.3) na = «„_!_«•
This follows immediately from (8.2.4) and formula (iii) in (8.3.1). ?
(8.3.4) Corollary (range of the spectrum) The spectrum (as a set of
rational numbers) is contained in the interval (—1, ri)
(8.3.5) Sp{f) С (-1, и). О
Moreover,
(8.3.6) F'H^X^ С) = #-(*<», С)
8 Spectrum of a hypersurface singularity
and
(8.3.7) V>~1 D.JffmDV"-\
where „?f@) is the Brieskorn lattice and V- is the root filtration.
119
Proof By Malgrange's theorem A.6.2.1) it follows from F.5.6) that
j^o) c F>-i; and hence GrPvj?m = о for /3 =s -1 and, consequently,
Fn+l = 0. Hence, na = 0 for a =s -1, i.e. Sp(f) С (-1, oo). The sym-
symmetry (8.3.3) implies that Sp(f) С (-1, n). It follows that GrpFHk = 0 for
p < 0 (and, moreover, GrpHx = 0 for p < 1, if X = 1), i.e. F°H = H.
Furthermore, Gr^yM^ = Cp for /3 ^ n — 1 (otherwise there exists a
spectral number 3= n), i.e. leading parts of geometrical sections generate Cp
for p s* n - 1. It follows that .^<°» D Vn~l. ¦
8.4 The spectra of a quasihomogeneous and a semiquasihomogeneous
singularity
Let f{x) = Y.cn,xm, m = (m0, ..., mn), xm =x™°- ... •**", be a quasi-
homogeneous singularity of degree 1 with variables of weights wt^- = wj.
The (quasi)degree of monomials xm
{oo},
(8.4.1)
defines the order function (valuation)
v: 6>Xfi = С{*ь, • • •, *„} -+ Q
where for a series g(*) = ^fcm;tm we set
(8.4.2) v{g) = min {v(xm): bm ф 0}.
The inclusion QJ^1 С @x,o-№olxo) Л ... Л (ск„/;сл) defines an order
function on the space of germs of differential forms Q J+o', which we also
denote by v: ?2J+o —*• Q>_i U {схэ}, and where for a form a> =
g(x) dxo Л ... Л йх„, g € C{x0, ..., х„}, we set (with a shift of — 1)
(8.4.3) v(a>) = v(g-xo- ... -х„)-\.
In particular, for a monomial com = xm dx
(8.4.4)
7=0
df
The order function v defines a decreasing filtration on Qx*q ¦ We want to
transfer the order function and this filtration to the quotient Ж^ =
Qx+d /df Л d?2J~0'. In what follows we'll meet this situation of coming to

120
II Limit MHSs
a quotient space more than once. Therefore we make the following general
remark.
(8.4.5) On the order function and the filtration on a quotient. Assume that
on a group (module, ring etc.) M there is an order function a:
M —> Q^o U {oo} satisfying the condition
a(m\ + mi) ^ min{a(mi), а(тг)}, Vm\, тг € M,
where the equality holds if a(m\) ф а(тг).
With an order function a one associates a decreasing Q-filtration V\
VPM ={meM: a(m) s= ?},
with V>$M defined analogously. The joined object Gr'vM has compo-
components Gr®vM = V^M/V>^M. The order function a may be recovered by
the filtration V.
a(m) = max {j3: meVp}.
Now let N С M be a subgroup and M — M/N its quotient group. We
can define a function a: M —> Qs0 U {00} on M,
a([m]) — max {a(m + n), n € N}.
Then we can consider two filtrations on M:
(i) a filtration V- associated with a,
Щ~М) = {[т], Щт]) s* ?} and
(ii) a quotient filtration V\
fp(M) = (VPM + N)/N с M/N = ~M.
It is easy to verify the following proposition.
Proposition. V' = V' Therefore in the following we denote by a and V'
the order function and the filtration on a quotient.
We now return to the Newton order function v on QJ^1. Let v:
.^f@) _> q30 и {00} be the corresponding order function on the Brieskorn
lattice .^f<0). We showed in A.5.5.4) that for monomial forms com = xm dx
we have td,[com] = (a(m) - l)[<om] = v(wm)[(om], i.e. [сот] е CHu)) and
(8.4.6) a([(om]) = v(wm) = v([Q)m]),
where a is the order function defined by the root decomposition *Ж =
©Ca in F.3.1). It follows that
(8.4.7) a(co) = v(co) for all со е
8 Spectrum of a hypersurface singularity
This implies the following proposition:
121
(8.4.8) Proposition If {xm}, m 6 A, \A\ = /л, is a monomial basis of the
Jacobian algebra Q/ = б x,o/Jf, then the forms com = xm dx form a basis
of the б 5,о -module J?m, and their orders
a(com) = v(com) = a(m) - 1
form the set of spectral numbers,
(8.4.9)
W) = 5>(а>„)).
meA
For the spectrum as a fractional Laurent polynomial we have the elegant
formula:
(8.4.10) Гр(Л=Цг^]
(see [S3] and (8.5.8)).
(8.4.11) Example Let f(x, y) = x3 + y4 be the singularity of type Ев-
Then wtc = \, wty = |. The following monomials xm form a basis of Qf,
and for the values of v(ct)m) = a(m) — 1 we have
xm: 1 у x у2 xy xy2
v(com): -f2 -± -± ± ± Jl
By the formula (8.4.10) we have
= f(r5/12 + Г2/12 + Г1/12 + f1/12 + /2/12 + ?5/12).
As shown by Steenbrink [S2], GrwkHn(Xx) = 0 for к ^ n, n + 1 and <
(8.4.12)
{com: и - p - К v((«m) < и - /?} is a basis of GrpFGrwnHn{Xao\
{com:v(com) = n - p] is a basis of Gr?Gr?.,tf"(Ar00).
In particular, for quasihomogeneous singularities all second numbers / of
spectral pairs are equal to n.
A singularity f(x) is called semiquasihomogeneous with weights
wo, ..., vvn, if f{x) = fo(x) + g(x), where fo(x) is an isolated quasi-
homogeneous singularity of degree deg/0 = v(f0) — 1, and v(g)> 1. A
semiquasihomogeneous singularity is a ,M-const deformation of its quasi-
homogeneous part fo(x). By Varchenko's theorem (see (8.9.11) below) the
spectrum is constant in ^-const deformations. Thus, for a semiquasi-
semiquasihomogeneous singularity f(x) the spectra Sp(f) and Sp(fo) coincide.

122
II Limit MHSs
8.5 Calculation of the spectrum of an isolated singularity in terms of a
Newton diagram
(8.5.1) Let / = ?cmxm € 6 = ^c-',o = <?{xo, •••,*»} be a germ of
an isolated singularity, supp/ = {m e Nq+1: cm Ф 0} be its support,
Г+(/) be the Newton polyhedron, i.e. a convex hull of
U mesuppfim + R"+1) С U"+l, and T(f) be the Newton boundary (or the
Newton diagram), i.e. a union of compact faces a of T+(f). We assume
that /is Г-non-degenerate (non-degenerate with respect to Г(/)), i.e. for
each compact face a of Г(/) the polynomials dfo /dx0, ..., dfa /дх„ have
no common zeroes on (C\{0})"+I. Here fo =Y.mecone@,a)Cmxm,
cone @, a) is a cone over the face a with a vertex at the point 0. Moreover,
we assume that Г is convenient, i.e. Г intersects all coordinate axes or f(x)
contains monomials xj', j = 0, ..., n. This doesn't affect the generality,
because /has an isolated singularity and for large mt the germs /and
/ + J2j=ox7J are -^-equivalent.
(8.5.2) Let us consider a homogeneous function h:
ри+1
ь h{la) =
Xh(a), such that /г(Г) = 1. Г defines a decreasing nitration on the ring
@ = ^'c»+i>0
(8.5.3) y/ •"<«' = { g(x) e C-: A(supp g) > a},
called the Newton filtration, and correspondingly Л'>а@ and Gr"r =
Жа/Ж>а. The nitration .///-- defines an order function v:
@ —> Qao U {oo}, where the Newton order or the Newton degree of an
element g € & is given by
(8.5.4) i<S) = max {a|g еЛ''в}.
The Newton filtration Ж' induces an order function v of a quotient
filtration on the Jacobian algebra Qf = &/J/.
The filtration.// ' and the order v on v induce the Newton filtration and
the Newton order on Q^+o', where v: Q^1 —> <Q> _j U {oo} is defined by
(8.5.5) v{(o) — v(g-x0 ... xn) - 1 for со - g(x)dx0 Л ... Л дх„.
Correspondingly one defines the quotient filtration .///- and the Newton
order v:.jy@)->?&>_, U {oo} on the quotient J?f@) = Q^+0V
(8.5.6)
v(M) = max {^G7): [77] = [a,]},
a],
and analogously for Qf = .Ж^/.Ж'-ч = QJ+0'/d/ Л Q%.
With a filtration F" on the vector space H there is associated the
Poincare polynomial
8 Spectrum of a hypersurface singularity
123
(8-5.7)
where ga - dim Fa/F>a.
M. Saito [Sa6] proved Steenbrinks's conjecture [S3] on the calculation
of the spectrum in terms of the Newton filtration
(8.5.8) Theorem For a Г-non-degenerate function / with an isolated
singularity the root filtration V~ and the Newton filtration Ж' on the
Brieskorn lattice .Я?@) coincide and therefore
(8.5.9)
Sp(/) = ? ta> - pQ,,v.(t) = Pa,,. 1 @-
1=1
Varchenko and Khovanskii [V-Kh] gave an elementary proof of this
theorem.
(8.5.10) Steenbrink [S3] obtained, using results of Kouchnirenko [Ко], the
following formula for the calculation of paj4..i @ in terms of the Newton
diagram. Let a be a face of T(f). We define Aa to be the C-algebra
generated by the monomials xm for m € cone@, a). The Newton filtration
on О induces a filtration on Aa с S:'. Let pAa (/) be the Poincare series of
Aa associated with this filtration. Then
(8.5.11)
a
where the summation is taken over all faces a of T(f), and k(o) is the
dimension of the minimal coordinate plane containing a.
The function h from (8.5.2) is linear on cone@, a), and if this cone is
'basis', i.e. it is generated by a part of a basis of the lattice of monomials,
then Aa is isomorphic to a polynomial algebra, Aa = C[y), ..., ук],
graded by the condition wtyt = w, = v(yi), i — 1, ..., &. In this case the
Poincare series is well known ([AGV]),
(8.5.12) Рл.(*)= 1/A-t»')... (I-I)**,
(8.5.13) Example Let f(x, y) - xr + x2y2 + ays, афО, г, s э= 4, r or
s > 4. This is the singularity of type T2tr,s [AGV]. In this case T(f) consists
of two segments Оц and 023 joining points O\ and аз, and o2 and аз
where O\ = (r, 0), o2 = @, s), a3 = B, 2). There are five faces in
П/): °\> °2, Oi, On and а2з. For them the number k(o) equals 1, 1,
2, 2 and 2, respectively. The cone cone @, Оц) (respectively

124
II Limit MHSs
cone@, cr23)) is generated by m\ =A,0) and тг = A, 1), and it is
'basis'. Since v{x) = \/r, v{xy) = |, we have by (8.5.12) that p^ =
A _^i/2)-iA_^A)-i The cones COne@, a,), cone@, a2) 'and
cone@, 03) are generated by A, 0), @, 1) and A, 1), and also are 'basis'.
We have рАа> =(\-&Т\ Рл„2 =(\-txly\ pAai = (l-tx'2T
Theorem (8.5.8) and formula (8.5.11) give that Sp(f) is equal to
Pq,,.i H) = О - 02(l - '1/2Г'A - tx/ryX
Tl
- A - 0A - tl/r)~l ~ A - 0A - tl/')~l
(8.5.14) Example Let f(x, y, z) = x? + y* + zr + axyz, а ф 0, p~x +
q~l + r~l < 1, be a singularity of type T p,q,r- In this case T(f) consists of
three triangles cr]24, огъл, оъи defined by three of the four points
ox{p, 0, 0), ст2@, q, 0), ст3@, 0, r) and ст4A, I, 1). There are 13 faces in
Ц/): СП, СГ2, СГ3, СГ4, O\2, О23, СГ31, СГ14, 024, СГ34> СГ124, 0234, СГ314. For 04
and for the last six of these faces k(a) = 3. The cones over all the faces are
'basis'. Hence, by (8.5.12) we have
Рл.п = О -
РА„Н = A -
etc.
Calculating pah.r(t) using the formula (8.5.11), we obtain by the
theorem (8.5.8)
Sp(f) =
ti/p)
(8.5.15) Example Let f(x, y, z) = *8 + yg + z8 + x2y2z2.
The same calculations as in (8.5.14) will give
8 Spectrum of a hypersurface singularity
-w + зг3/8 + 6r2/8 + 9r'/8 +13
25tx/2 + 24t5/* + 21 tbli + Ш7/8 +
125
In particular, we obtain that fj. = 215 and the characteristic polynomial
of the monodromy is (f8 — \J1(t - \)~x.
See also Danilov's paper [Da] about the calculation of the Hodge
numbers hpx'q in terms of the Newton diagram.
(8.5.16) The calculation of the negative part of spectrum. On the Brieskorn
lattice .Ж'@) = Q?+o' /d/ Л dQ?"„' as on a quotient, the Newton order v is
defined by formula (8.5.6)
v([a>]) = max {v(o) + rj), rj e d/ Л dQ$r0'}.
The function v satisfies the property v(a> + 77) 5= min {v(a>), v(?7)} and
v((o + fj)~ Ч60), if viw) ^ vG/)- Hence for со such that v(a>) < min {v{rf),
r/ € d/ Л dQ?"„'}, we have
(8.5.17) ' v([o)]) = v(a>).
Let
(8.5.18)
s = min{v(cy), w €d/AdQJ0}
df '
= miniv(-—xo- ... ¦ х„) - 1,7 = 0, ..., и I.
L \axj / )
It is obvious that s>0 (because v(df/dxj-Xj) = v(f) — 1) and that
min {v(co), w € d/ Л dQ?'}. We then obtain:
(8.5.19) Proposition If v(w)<s, then
(8.5.20) Let у4 С Mq+' be a set of exponents m of monomials xm for 1
which
v(A:mxo-... xn)-Ks
and correspondingly /4o с A be a set of exponents of the underdiagram
monomials

126 II Limit MHSs
A0={m:v(xmx0- ... -хя)-1 « s].
Ao is a set of those m for which m + A, ..., 1) doesn't belong to the
interior of the Newton polyhedron.
Then for the forms com =xm due, m € A, we have
a(com) = v(com) = v(x"'x0- ... -х„) - 1.
Now note that the images of monomials хт,т€А,ате linear independent
in the Jacobian algebra Q/ and hence forms com, m € A, are linear
independent modd/AQJ0, i.e. they are linear independent in
Qf=.Mm/.J9?<--lK Indeed^ by definition of the set A for any linear
combination Х)телс„,хт we have v((Y,meACmxm)x0 ...xn)—l<s and
Y,m4ACmxm dx = J2>»?ACma)m ? d/ Л Q"x0 by the definition ofs.
Moreover, any form со = g(x)dx, g(x) = Y^n,cmxm = Y
, can be written in the form
CO = ^ CmXm + T),
meA
where v(jj) =* s and, consequently, a([rj]) = v([rj]) s» v(rj) s= s. We obtain:
(8.5.21) Corollary The images of forms com, m € A, form a basis in the
space ®-\<a<sGrav.Jtf@), and also in ©_i<a<JG^Q/, and their orders
a(co), m € A, are exactly the spectral numbers in the interval (—1, s). In
particular, a(com), m ? Ao, is exactly the part of spectrum in (-1, 0] and
hence
(8.5.22)
-\<a<0
Moreover, the least spectral number is given by
(8.5.23)
(8.5.24) Corollary (8.5.21) enables us to calculate the spectrum completely
in the cases of curves (n = 1) and surfaces (n = 2):
A) If n = 1, /: (C2, 0) -> (C, 0), then Sp(/) С (-1,1) and Sp(/) is
symmetrical wrt the point 0. Knowing the part of Sp(/) in (—1, 0] we
know Sp(/) completely by symmetry.
B) If и = 2, /: (С3, 0) -»(С, 0), then Sp(/) с (-1, 2) and Sp(/) is
symmetrical wrt the point |. Knowing the spectrum in the interval
(—1, 0], we find it in [1, 2) by symmetry. For the other eigenvalues A
of the monodromy the numbers a =—A/2m) log A give the part of
spectrum in the interval @, 1).
8 Spectrum of a hypersurface singularity 127
8.6 Calculation of the geometric genus of a hypersurface singularity in
terms of the spectrum
Recall that the geometric genus pg of a normal isolated singularity (Y, 0)
of dimension n is defined by means of its resolution я: Y —> Y,
(8.6.1) pg = dim R"~] л^-у,
if n ss 2. For curves (n — 1) pg = д is the й-invariant of the singularity,
6
(8.6.2) M. Saito's theorem [Sa3] For an isolated hypersurface singularity
(У, 0) с (C"+1, 0) with equation/(x) = 0
Pg = dimF"H"(XO0,C)=
i.e. the genus is equal to the degree of the non-positive part of spectrum.
8.7 Spectrum of the join of isolated singularities
Let /(x) € C{jco, ..., х„} and g(y) ? C{ya, ..., ym} be isolated singula-
singularities of n + 1 and m + 1 variables, respectively.
(8.7.1) Definition The join (or the direct sum) of singularities /and g is
the singularity f ® g in л + ю + 2 variables defined by the function
/(*) + g(y) € C{x0, ..., х„; yo, ..., ym}.
This singularity is also isolated. Let X-f -> S, Xfx, J^ etc. be the
Milnor fibration, the canonical fibre, the Brieskora lattice etc. for the
singularity / (and analogously for g and / ф g).
Singularities of the form /®g and the results about them are called
Sebastiani-Thom singularities and 'Sebastiani—Thorn' results, respectively
because Sebastiani and Thom [Se-T] proved that the Milnor fibre x?^s
has the homotopy type of the join of the Milnor fibres of / and g,
XfJ>s ^X^* Xg,, and in particular
(8.7.2) Hn+m+l(X^s, С) ~ Hn(Xi, С) ® Я"ЧЛ^, С)
and (xf®s = /г?-/*8. They also calculated the monodromy of the singularity
/eg, T'®s = Tf ®Tg (See also [AGV]).
(8.7.3) Theorem If {a,} and {/?,} are the spectra of isolated singularities
f(x) and g(x\ then
(8.7.4) {a,+ft + l},/=l,...,//,y=l ц*,
is the spectrum of the singularity/ Ф g, or

128 II Limit MHSs
(8.7.5) Sp(/ ®g) = tSp(f)Sp(g),
if we consider the spectrum as an element o
/~l/m
].
e c p ^
This formula isj>f a more symmetrical form Sp(/ ® g) - Sp(/)Sp(g)
for the spectrum Sp(/) = tSp(f) according to M. Saito. This theorem was
conjectured by Steenbrink [S3] and proved by Varchenko [V3] and Scherk
and Steenbrink [Sc-S] for the case of isolated singularities. M. Saito proved
it for the general case [Sa9].
(8.7.6) The spectrum of the zero-dimensional singularity Ah g{y)-y2,
consists of one number -1/2, Sp(g) = (-l/2) or Sp(g) = Г1'2. We
obtain the useful consequence:
(8.7.7) Corollary If a singularity/^) has spectrum {a,}, then the singu-
singularity f(x) + y2 has spectrum {a,- + 1/2}, i.e. adding a square of a new
variable leads to a shift of the spectrum by 1/2.
Actually Scherk and Steenbrink [Sc-S] expressed the limit MHS on
Hf®g = #«+'«+i(x?fg, C) in terms of the MHS on H' = #"(*?,, Q
and Hg - Hm{Xgs, С). They proved the following theorem:
(8.7.8) Theorem Let (Hf, W., F) and (#*, W., F) be limit MHSs on
#/ and Hg and let Hf = @хн{ = 9-ка«о^, Hg = Ф-кр^оЩ be
root decompositions of the monodromy operators on H^ and Hg. Then the
MHS on Hf®g = H= Hf <g>€ Hg is the join of the MHS on Hf and Hg
in the following sense:
(i) the weight filtration W. on H
WkH =
\ а+р-ф-1, either афО or
(ii) the Hodge filtration F- on H
FpH =
8 Spectrum of a hypersurface singularity
129
As a consequence of this theorem we obtain the formula expressing
Hodge numbers hpx'4 for the singularity/ © g by means of Hodge numbers
for the singularities/and g.
8.8 Spectra of simple, uni- and bimodal singularities
The methods of calculation of spectra considered in (8.4.8), (8.5.8),
(8.5.10) and (8.7.3) enable us to calculate spectra for a wide class of
singularities. In particular, there are tables of spectra of simple, uni- and
bimodal singularities [Go]. We give such a table in (8.8.1) for the case of
simple and unimodal singularities /: (C3, 0) —> (C, 0), n = 2. In this case
the spectra are contained in the interval (—1, 2) and are symmetric wrt the
point 1/2. Table (8.8.1) show the singularity notation of according to
Arnold and the products (Na\, ..., Na^) of spectra {a,} and the num-
number N.
(8.8.1) Table
Type of
singularity
N (Nau...,Na/l)
Л + 1
Еь
En
E%
TP,4,r
En
Е\з
E\4
Z\\
Zn
Z.3
Qio
Qu
On
w\l
Sn
Sn
U,-,
12
18
30
pqr
42
30
24
30
22
18
24
18
15
20
16
16
13
12
A,2,..., /i)
A,3,..., Ifi - 3,|<-2)
A,4,5,7,8,11)
A,5,7,9,11,13,17)
A,7,11,13, 17, 19,23,29)
@, pqr, kqr, l-pr, mpq)
0<k< p,0< Kq,0<m<r
(-1,5,11, 13, 17, 19,23,25,29,31,37,43)
(-1, 3, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 31)
(-1,2,5,7,8, 10, 11, 13, 14, 16, 17,19,22,25)
(-1,5,7,11, 13, 15,17,19,23,25,31)
(-1,3,5,7,9,11,11,13,15,17,19,23)
(-1,2,4,5,7,8,9, 10,11, 13, 14, 16,19)
(-1,5,7,8, 11, 13,16,17,19,25)
(-1,3,5,6,7,9,11,12,13,15,19)
(-1,2,4,5,5,7,8,10,10,11,13,16)
(-1,3,4,7,8,9,11,12,13,16,17,21)
(-1,2, 3, 5, 6, 7, 8, 9,10, 11, 13,14,17)
(-1,3,4,5,7,8,9,11,12,13,17)
(-1,2,3,4,5,6,7,8,9,10,11,14)
(-1,2,3,3,5,6,6,7,9,9,10,13)

130 II Limit MHSs
This table contains, in particular, the spectra of simple elliptic singularities
Еь — Р% = ?3,3,3, Ej = Xg = ?2,4,4 ai*d E$ = J\a = 72,3,6-
Now we proceed to the investigation of variations of Brieskorn lattices
and MHSs in families of hypersurface singularities. The next chapter is
devoted to continuous invariants (moduli), but here we'll consider the ,
behaviour of discrete invariants under deformations.
8.9 Semicontinuity of the spectrum. Stability of the spectrum for ц-const
deformations
Let /: (C+1, 0) —»(C, 0) be a germ of a holomorphic function. Assume ,
that 0 is an isolated singularity. Let F\\ C"+1 X Y —» С be a deformation
of / i.e. a family of functions fy — F\(., y): C+1 X {y} —» С parame-
parametrized by Y. Let
(8.9.1) F:i'->5X Y, F(x, y) = (F, (x, у), у)
be good representative of the deformation of the corresponding hyper-
hypersurface singularity (/"'@), 0) = (X, 0), where S={teC: \t\<6}, Y =
{уеСт:\у\<д'}, ./' = {(д:^)бС"+1ХУ: |x|<e, |F,(x, jO|<<5},
0 < 6, 6' < 1. It appears uncertain how the spectra of singularities vary
under deformations. ;
(8.9.2) ArnoPd was the first to discover the importance of the spectrum for
deformation theory [Ar]. He conjectured that the spectrum behaves semi-
continuously under deformation in the following sense: if a singularity P
adjoins a (simpler) singularity P' with /л' <ц and a\ =? a2 *s ... =s aM,
a\ =? ... =? a'p! are their ordered spectra, then a.k *s a\. From this (and
from the symmetry of the spectrum) it follows that when /г' = /г — 1 the
spectrum of P' 'separates' the spectrum of P: a\ =s a\ =s 3
<Хг =? а'г «s ... =? a^_i =s a^. We formulate the property of the semiconti-
semicontinuity of the spectra in terms of the notion of the semicontinuity domain
[S5].
(8.9.3) Definition The degree of the spectrum Sp(f)
singularity f relative to a subset В С IR is the number
degB Sp(/) =
of spectral numbers contained in В.
In particular, if В = R, then
of a
8 Spectrum of a hypersurface singularity 131
(8.9.4) degB Sp(f) = ^ na = fi is the Milnor number,
and if В is a half-open interval of the form В = (и — p — 1, n — p], then
(8.9.5) degBSp(f)= Y, na = dimGrpFH'l(Xx,C).
n—p—l <a*in—p
When p = n, В = (-1, 0], we obtain by M. Saito's theorem (8.6.2) the
geometric genus of the singularity: deg(_i,o] Sp(/) = pg. It is known that
the geometric genus is semicontinuous under deformations of the singular-
singularity [E]. This is a particular case of the semicontinuity of the spectrum.
Assume that there are several singular points x\, ...,xr in the fibre
F~\z) = .?\, z — (t, y), of a deformation F: .?' -> Z = S X Y of a
singularity (X, 0). Denote by Sp(/Z) = ?,r=isP(/r> *<) the sum of all the
spectra of the function fy: C+1 X {y} —» С at the critical points
x\, ...,xr having the same critical value t. We define the degree
(8.9.6)
degB Sp(fz) =
Sp(fy, xt).
(8.9.7) Definition A subset В С R is called a semicontinuity domain for
deformations of isolated singularities, if for every deformation F, the
function z i—> degB Sp(fz) is upper semicontinuous on Z,
degB Sp{f) > degB Sp(fz)
on some neighborhood of the point 0 € Z, f — /o.
(8.9.8) Theorem [S5] Every half-interval В = (a, a + 1] of length 1 is a
semicontinuity domain for deformations of isolated hypersurface singula-
singularities.
The theorem is first proved for the case В = (n - p - 1, n — p], i.e. the
semicontinuity of the function &xmGrpFHn{XO0, C) is proved. Then the
theorem is proved for any a by means of Varchenko's trick [V8] using the
relation between the spectra of the functions /(x) and/(x) + z4,
(8.9.9) Sp(f(x) + z«) = {a+ k/q\a € Sp</), * = 1, ..., q - 1},
(see (8.7.3) and (8.8.1)).
(8.9.10) Arnol'd conjectured that any half-line В = (—oo, a] is a semi-
continuity domain. Theorem (8.9.8) implies the validity of this conjecture.
Varchenko's conjecture which is stronger than theorem (8.9.8), is that any
open interval В = (а, а + 1) is a semicontinuity domain. He proved this
[V8] for low-weight deformations of quasi-homogeneous singularities.

132
II Limit MHSs
Theorem (8.9.8) also implies the low semicontinuity of the smallest
spectrum number amjn (or, equivalently, of the complex singularity index),
which was conjectured by Malgrange [Ml]. Finally theorem (8.9.8) implies
a new proof of the following theorem of Varenchenko [V5]:
(8.9.11) Theorem The spectrum is constant under a deformation of iso-
isolated hypersurface singularities with constant Milnor number (= under a
/г -const deformation).
Indeed, ju = J2na remains constant and degB Sp(f) = ?aes«a,
В = (a, a + 1] and degB Sp(f) can only jump. Hence, degB Sp(f) remains
constant for any a. This is only possible if the spectrum remains constant.
8.10 Spectrum of a non-isolated singularity
The spectrum Sp(f) of an isolated singularity /: (C"+1, 0) -> (C, 0) was
defined in (8.1.4) by the following data. Given a vector space
H = Яи(^0о, С) of dimension ft, an operator у = Ts acts on H with
eigenvalues X\, ..., Xf, which are roots of unity, and there is a decreasing
y-stable filtration F- (the limit Hodge filtration). The operator у also acts
on GrF(H) - ®pGrpF(H) = ©pF'pIF'p+i. By definition the spectrum
Sp(f) is associated with the triple (#, y, F), Sp(f) = Sp(H, y, F-), and
it is a set of logarithms of the eigenvalues yj, cij = -(l/2m)logXj,
j = 1, ..., ft, which are normalized according to the distribution of Xj in
the decomposition ®pFp/Fp+] by condition (8.1.3). The choice of the
normalization (8.1.3) is motivated by the definition of the filtration F' on
H. In addition this choice depends on n. We can consider a more natural
normalization:
(8.10.1) a = — logX and [a] = p, i.e. p < a<p + 1,
if an eigenvalue X corresponds to Fp/Fp+l.
Let Sp' = Sp'(H, y, F') be the spectrum obtained according to normal-
normalization (8.10.1). Just as in (8.1.4) and (8.1.6) we can think of Sp' in three
ways:
(8.10.2) as a set of spectral numbers Sp' = {a\, ..., a^};
as a 'divisor' Sp' = J2<xna<a) € Z(Q);
as a Laurent polynomial Sp' = ^anata € Z[fI/m, t~^m].
The way in which the spectrum Sp' is considered is always obvious from
the context.
In the group Z(Q), respectively in the ring Z[f1/m, Гх/т], there is an
8 Spectrum of a hypersurface singularity 133
involution / defined by (a) >-> (-a), respectively, ta ь-> ra. Under the
change a i-> n - a the condition p «s a <p + 1 (8.10.1) transforms into
condition A0.1.3) or equivalently n — p—\<n — a^n-p. Thus, if we
define
(8.10.3) Spn(H, y, F) = tn/(Sp'(H, y, F-)),
we obtain that for an isolated singularity
Sp(J) = Spn{H, y, F-)
where H ^{"(X^, С), у = Ts, and F- is the limit Hodge filtration. We
recall that Sp(f) = tSp(f) is the spectrum according to M. Saito (8.1.9).
Now let/: (Cn+1, 0) ->• (C, 0) be any, not necessarily isolated, singular-
singularity, /: X —» S be the Milnor fibration, which is defined in the same way as
in the case of isolated singularities, and Xx be the canonical fibre. Then <i
there is an MHS \N) on the cohomology #*(ЛГоо) which is invariant wrt U
the semis imple pa"rtof the monodromy у = Ts. '
The spectrum of a singularity / is defined [S6] by the formula
(8.10.4) Sp(f) =
*=0
+ • • •
where Hk(X0o) are reduced cohomologies.
In the case of isolated singularities this definition coincides with the
previous one.
(8.10.5) Examples
A) A^. f: (C3, 0) -»(C, 0), f(x, y, z) = xy.
In this case Sp(f) = -A) or Sp(f) = -t since the fibre X^ is homeo-
morphic to the affine variety xy = 1 in C3, i.e. to C* X C, and T — id.
B) Г,».,,...,». /: (С3, 0) -+ (С, 0), f(x, у, z) = xyz.
In this case Хж ~ {xyz = 1} ~ С* Х С*, Т = id and the MHS on
HXXoo) is pure of type (j, /), i = 0, 1,2. Hence we obtain that Sp(f) =
@)-2-(l).
These examples show that unlike what happens with isolated singula-
singularities the spectrum Sp(f) need not be symmetric wrt the point (n — l)/2.
(8.10.6) Remark The Sebastiani-Thorn formula (8.7.5) Jor the join of
singularities, which Jor thejpectrum according M. Saito Sp(f) = t-Sp(f)
reads Sp(f ® g) = Sp(f)-Sp(g), was also proved by M. Saito [Sa9] in the

134
II Limit MHSs
case of non-isolated singularities. This formula also remains valid in the
rase of a zero function in к variables, if for such a function we set
Sp(g) = (—\)ktk. In particular, for к = 1 we obtain
(8.10.7) Sp(/-©0) = -fSp(A
i.e. if/: (C, 0) -»(C, 0) has the spectrum Sp(f) = X>«-(a), then the
function g: (C+1, 0) -* (C, 0), g(x0, xu ..., xn) = /(*,, ..., х„) has the
spectrum Sp(g) = -J2na-(a + 1).
8.11 Relation between the spectrum of a singularity with a one-
dimensional critical set and spectra of isolated singularities of its Iomdin
series
Let/: (C"+1, 0) —¦> (C, 0) be a singularity with a one-dimensional critical
locus 2 = Singf = Singf~l@). Let 2 = 2i U ... U 2r be decomposition
into the union of irreducible components. Let m, be the multiplicity of the
one-dimensional singularity B,, 0) There are two ways to relate / to a
deformation of isolated singularities. Firstly, /can be considered as a limit
(degeneration) of one-dimensional families fc, e —> 0, of isolated singula-
singularities as follows. Let / be a general linear form (coordinate) on C+1. We
consider a deformation
(8.11.1) /E
of the singularity / Then for all к sufficiently large and 0 < |e| <§: 1 the
functions fe have an isolated singularity at 0. We call the series (8.11.1) of
isolated singularities /* (with variable k) the Yomdin series, since Yomdin
[Y] obtained the relation for Milnor numbers of singularities /and/t
(8.11.2) /Kfk) = M(f)+k-mC?),
where fj.(f) = {in(f) — Mn-\(f) is the Milnor number of the non-isolated
singularity /which, by (8.10.4) is equal to the degree of its spectrum
ju(f) = deg Sp(f) (as in the case of isolated singularities), and тиB) is the
multiplicity of the singularity B, 0) Steenbrink [S6] generalized this result
and obtained a formula relating the spectra of singularities / and /*.
Steenbrink proved this formula in different specific cases, and M. Saito
[Sa9] proved it in the general case.
(8.11.3) The second way to relate /to deformations of isolated singula-
singularities is to consider / as itself a deformation / = fy of isolated singula-
singularities fy of one less dimension. Let us take a general linear form / on
C+1 to be one of the coordinate functions, which we denote by
y. Let (jci, ..., х„, у) be coordinates on C+1, f(xit ..., х„, у) =
8 Spectrum of a hypersurface singularity 135
fy(x\, ..., х„). We can consider a function /: C+1 —» С as a family of
functions fy. С X {у} -* С. The hyperplane С" = С X {у} intersects
the curve 2 transversely in r points P,¦ = 2,- П C", i — 1, ..., r. For у ф 0
we get r isolated hypersurface singularities:
(8.11.4) gi: (C;, Pi) -> (C, 0), gi =fy,t =/|C».
The /л-class of the singularity g, depends only on / (for general I) and is
called the transverse type of the singularity f along 2,-.
СЭ/
Let2* = 2Д{0}. We can (locally) consider/: C"+1\(C X {0}) -> (C, 0)
as r ^-const one-parameter deformations fy: U[=i(C", Pi) —»(C, 0) para-
parametrized by the curves 2*, i = 1, ..., r (or by C\{0} Э y).
In the (8.11.5)—(8.11.8), we will make a small digression about one
important general question: Deligne's method of vanishing cycles.
Deligne 's sheaves of vanishing cycles
(8.11.5) Let/: X —> S be a morphism of an analytic space Xonto the unit
disk and let its restriction /: X' —> S' be topologically locally trivial over
S' = 5'\{0} and let there exist a retraction r: X —> Xq onto the zero fibre,
compatible with the trivialization. By Thorn's isotopy theorem any proper
morphism / is of such a topological type in a neighborhood of the zero
fibre (see [Di3], Ch. 1). The situation is the same for the case in which /is
the Milnor fibration of a germ of a holomorphic function. Let X, = /~'@
and let r,: X, —» Xq be the restriction of the retraction onto the fibre X,.
The local triviality of / over S' defines the monodromy transformation
h: X, —> X,. From the topological point of view the map /: X —> S is
reconstructed by the quadruple (Xt, Xo, h, rt): first we get the fibration
/': Xх —> Sl over the circle S1 ghung Xt X [0, 1] by means of h, and
then we get las a cone cone(X* ^* Xo) of the map r1 defined by the

136
II Limit MHSs
retraction r,. The map /: X —> S corresponds to the map of cones
cone(Z' -^ Xo) -> cone (S1 -> pt) = S.
(8.11.6) Deligne's method of vanishing cycles consists of obtaining infor-
information about the difference between the cohomologies of Xo and X, by
means of the Leray spectral sequence of the map r,: X, —> Xo, which for
the sheaf F = Cx, on X, reads as follows:
Ef - Я'(*о, **(»•/)*€*,) => Я'+*(Х„ С).
The sheaves <py-(C) = Rq(r,)*Cx (or V*) on Xo, which appear here, are
just Deligne's sheaves of vanishing cycles. By the definition of the direct
image the fibre of the sheaf <p'(C) at the point x ? Xo is equal to
Hg(XtrX, C), where XtyX is the Milnor fibre of the function /at the point x
(we take the intersection with a ball of small radius centered at the point x
etc.). The sheaves V* are cohomology sheaves of the complex
¦фу = Щг,)*Сх (the object of derived category). Finally to make the
construction independent of the choice of the fibre X, we have to pass to
the canonical fibre Xx (see B.4.5)): consider the diagram
(8.11.7) J0Cl t- XOO = XXU = X'XU
'fi I
s Л и,
where U-^S' is the universal covering of the punctured disk 5". The
canonical fibre Xx has the homotopical type of fibres X, since Uis simply
connected. The space X has the homotopical type of Xq by virtue of the
retraction r. X —* Xq- So we can consider the map k: Xx —> X instead of
the map r,\ X, —* Xo- We now come to the final definition of the sheaves
of vanishing cycles.
(8.11.8) Definition The sheaves of nearby cycles ip4f(Cx) are the co-
cohomology sheaves of the complex
This complex of sheaves ip/(Cx) on Xo (or more exactly, the object of the
derived category DXo) is also called the sheaf of nearby cycles. The
sheaves of vanishing cycles (pqfCx are the cohomology sheaves of the
complex
<P/(C*) = сопе(Сл-0 -> ipf(Cx)),
the cone over the natural morphism CXo = i*Cx —> ip/(Cx).
Fibres of the sheaves 'фу'Сх at points x € Xo are the vanishing cohomo-
8 Spectrum of a hypersurface singularity
137
logy, i.e. the cohomology Hq(Xt,x, C) of the Milnor fibre at the point x,
and fibres of the sheaves q>4j-Cx are the reduced cohomology Hg(X,j, C).
According to the ideology of the 'new' homological algebra we can
change Cx by an arbitrary sheaf .T or a complex of sheaves on X and
obtain functors of nearby and vanishing cycles ip/, cpf. Dx —+ DXo. These
functors are used in the construction of perverse sheaves on X out of
sheaves on Xo and X' by 'gluing' [G-M].
We now return to singularities with a one-dimensional critical locus.
(8.11.9) Vertical and horizontal monodromies Let /: X —> S be the Milnor
fibration of a singularity /: (C+l, 0) —> (C, 0). There is a sheaf of vanish-
vanishing cycles cpf(Cx) on Xo- This is a complex of sheaves with the
constructable cohomology q>qf(Cx) = -J^q{cpf{?x)\ where (ру"(Сх) = 0
on Zo\2 for all q and (p}(Cx) = 0 on 2* = 2\{0} = X* U ... U 2* for
}
n—l, and the fibre q>"f~x{Cx)pl at a point P,- = I,- П
x{ C
is
H"~x{X{gi), C), the vanishing cohomology of the isolated singularity
gr- (C", Pi) -> (C, 0). Thus on 2* = 2Д{0} there are local systems
Hi = cpnf-l(CXl)p* with fibres H"-\X{gi), C) at points P,- 6 2*.
On the space H""i(X(gi), C) there are two monodromy transforma-
transformations: the monodromy Г, of the singularity g,: (C^, P;) —> (C, 0) corre-
corresponding to a circuit around the origin in the complex plane С with
coordinate t, which is called the horizontal monodromy; and the mono-
monodromy T,- of the local system Я,- on the curve 2* с Xo = /~'@) corre-
corresponding to a circuit around the origin in the complex plane С with
coordinate у (t = 0). The monodromies Г, and r,-, i = 1, ..., r, commute
with each other (since the group Л\{С* X С*) = Z ф Z is abelian).
(8.11.10) Let us consider one of the branches 2,-, i = 1, ..., r. Let ^, be
the Milnor number of the singularity gh Sp(gj) = YTjL\ta" be the spec-
spectrum of the singularity gh and A,,y- = е~2л^-7'"!' be the eigenvalues of Г,-.
Let k;,i, ..., Kij/ti be the eigenvalues of T/. Since Tt and r,- commute, we
can make the eigenvalues A,-,y and /c/,y- consistent, i.e. set them in one to one
correspondence in the following way. Choose a common eigenbasis
?;,i, •••> Zi,fi of the commuting semisimple operators (Г,M and (t,)s, and
associate to the eigenvalues A,,y- the eigenvalues /с,-,у- of the operator (t,)s
corresponding to the vector ey. Finally take a number j8,i7 such that
KUj = e-2*^", fa e @, 1].
(8.11.11) Theorem If/: (C+1, 0) -> (C, 0) is a singularity with a one-

138
II Limit MHSs
dimensional critical locus ? = (J|=i s< and fk=f + e-lk is the isolated
singularity of its Yomdin series, then for sufficiently large к 2= k0 the
spectra of/and fk are related by the formula
L
(8.11.12)
Sp{fk) - Sp{f) =
U
where /и, is the multiplicity of the branch B,, 0) and a,y and 0У are defined
in (8.11.10) by /г-const deformations of isolated singularities along
2,-\{0}, i = 1 r, which are defined by the function / The number k0
is determined by the discriminant of the map defined by (/, />
This theorem was conjectured by Steenbrink [S6] and proved by him for
n = 1 and for some other cases. M. Saito [Sa9] proved the conjecture in
general. He reformulated and proved the conjecture in the more general
context of his theory of mixed Hodge modules. D. Siersma [Si], proved by
topological methods a formula relating ^-functions of the singularities /
and fk, which can be considered as 'expBjri[formula (8.11.12)])'.
Ill
The period map of a /г-const deformation of an
isolated hypersurface singularity associated with
Brieskom lattices and MHSs
Until now we studied each singularity /: (C+1, 0) —>(C, 0) individually.
From now on we'll be concerned with families of singularities. Accord-
Accordingly to the general ideas of moduli spaces and period maps for non-
singular compact varieties in algebraic geometry, we have to consider the
set of all algebraic (or analytic) structures on the same underlying
topological variety. The period map is connected with the 'linearization' of
these objects: we replace the variety by its cohomology space, and the
algebraic variety structures by the spaces of differential forms on algebraic
varieties and HSs defined by them. In our case the underlying topological
object connected with a singularity / is its Milnor fibration /: X —> S and
the corresponding linear objects, viz. the cohomological Milnor fibration
H —> S' of rank fi = dim H"(X,, C) equal to the Milnor number, the
topological Gauss-Manin connection V on H, the monodromy transforma-
transformation T on the cohomology, the meromorphic connection ^#, the type of
which is defined by the Jordan structure of the monodromy T, and so on.
The complex structure of/is reflected in the Brieskom lattice J1'0' and in
its embedding in ^S, and in the corresponding MHS defined by this
embedding.
1 Gluing of Milnor fibrations and meromorphic connections of а /г-
const deformation of a singularity
Let /: (C+1, 0) -»(C, 0) be an isolated singularity. In chapters I and II
we associated to each such individual singularity the following objects: the
Milnor fibration f:X—*S; the corresponding cohomological (homologi-
cal) fibration H = \J tes'H"(Xh C) on S'; the vector space of multivalued
horizontal sections H of the fibration H, which can also be considered
as the canonical fibre Я;» = H"(XO0, C); the sheaf of sections
139

140
III The period map of a ц-const deformation
Ж = #®cs. <9s~ ^ J&qK(X'/S') of the fibration Я with connection V
D Rh hl sheaf
s
defined by this
= 3?^R(X/S)
fibration; the De Rham cohomology sheaf
M 3?^R(X/S) which is a natural extension of 3% on S; the
Brieskorn lattice :M(<S) which is another natural extension of 3% defined by
the geometric sections s[a>], where w runs through the set of (n + l)-forms
on X; the meromorphic connection on S, Ж = ^(~2) ®^s
&s\rx\ = Jg?@) ®^s ^[r1], ^# is a sheaf on S such that Ж\$' = 3%
and its fibre at the point 0 e S (denoted again by Ж) is the localization of
the fibre .%fQ0); Ж has the root decomposition Ж = ®aCa, where Ca is
isomorphic to the subspace Hx С Н corresponding to the eigenvalue
А = е~2яш, and the subspace Ca С Ж consists of the (single-valued);
sections s[A, a] = ta-tNA(t) of the sheaf Ж, where Л = A(t) e H, and
N = ~(\/2m)logTu; and finally, the canonical lattice (extension)
2! С Ж, 5% = У>~1Ж = фа>_|С„. Now we wish to consider the
corresponding objects for the singularities of a ^-const deformation of the
singularity /and to glue them to form a unified family.
1.1 Milnor fibrations
Let
A.1.1) F: (C+1 X Cm, 0) -» (C X C", 0),
t = Fi(x, _y), у — у, be a ^t-const deformation of the singularity /with a
smooth base space Cm э у = (y\, ¦¦¦, ym). To simplify the notation, we'll
often write t= F(x, y) instead of F\(x, y), if this does not lead to an
ambiguity. Thus, we have an ти-parametric family of isolated singularities
of functions
A.1.2) fy: (C+1 X {y}, 0) - (C, 0), fy(x) = F(x, y),
with constant Milnor number /j. for у in some neighborhood D С Ст of
the point 0, where fo(x) = F(x, 0) = f(x).
A.1.3) For every fixed у € D the singularity fy has the Milnor fibration
fy: X(y) —> S(y) = S, the corresponding cohomological fibration H(y) on
¦S" and so on. The letter у in parenthesis means that we are considering the
corresponding object for an individual singularity fy for fixed y. We want
to show, following Varchenko [VI, V5], how we can glue X(y), H(y) etc.
to form a uniform fibration over S' X D. The problem lies in the fact that
when constructing the Milnor fibrations X(y) —» ?(_у) of singularities fy
one chooses the radii 6 = 6(y) and e = e(y) for each у separately. Recall
1 Gluing of Milnor fibrations and meromorphic connections 141
the notation: В = Be = {x € C+1: |x| <e}, S = S6 = {t € C: \t\ <<5}
andD = Dv = {y ? Cm: \y\<v}-
Choose ?>0 such that the spheres dB?- intersect the hypersurface
f{x) = F(x, 0) = 0 transversely for s' € @, e] and that the point 0 is the
unique singular point of the function f(x) in Be. Choose r\ > 0 and д > 0
such that: A) for any yo € ?>7 the function fn(x) has no critical points
in B? distinct from 0; B) for any yo € Д, and any t e S^ the sphere
dBe intersects the hypersurface /л(х) = t transversely. Put 3S~ =
F~\SXD)D (B X D) and consider the map
A.1.4) F:3'->SXD.
Then over S' X D we have a smooth locally trivial fibration F':
Ж' -* S' X D. We get the family of fibrations:
A.1.5) .
НУ) = /,: ^"(У) -» 5 X M, where JF(y) = F~l(S X {у}), у е D.
To construct the Milnor fibration X(y) —> ¦S'a^j of a singularity
/,: C+1 X {j}, 0) -»• (C, 0), we have to take 0 < 6{y) < e(y) < 1 and
to consider the restriction of F onto the subset F^^) x {y})
П (BE(y) X {у}) П ^"(j) С Щу). The next theorem follows from the
paper of Le and Ramanujam [L-R].
A.1.6) Theorem The embeddings of fibres X(y), С ¦SS'ifit are homotopic
equivalences. ¦
Thus, from the homotopic (homological) point of view we can consider
F in A.1.4) as a family of Milnor fibrations X(y) —> S of singularities fy.
1.2 Cohomological fibration
The locally trivial fibration F'\ ЛГ' —> S' X ?) defines a flat cohomological
fibration over S' X Д
A.2.1) Я = U H"(JT(t,y), С),
where -%"(t,y) = F~x{t, y) is the fibre of the mapping F. Since
S ^"() i l h b
pg
is also the fibre of the fibration Ж(у) = F~l(S X {y}) at
the point t, we have from A.1.6) that the restriction of Я to <S" X {y},
A-2.2) Я|5-хМ=ЯМ
is the Milnor fibration of the singularity fy. One can say the same about the
dual homological fibration

142 HI The period map of a fi-const deformation
#*= U &№.», С).
(t,y)eS'xD
Denote by Я, respectively H(y\ the vector space of multivalued
horizontal sections of the fibration Я, respectively H(y). Obviously, Я and
H(y) are canonically isomorphic: the isomorphism H(y) -^ Я is realized
by extending sections from S' X {y} to S' X Z). The vector space Я is also
identified with the space of single-valued horizontal sections of the
fibration (e X id)~lH, where U-^>S' is the universal covering of S',
respectively U X De—» S' X Д and with the cohomology space
Я"(^оо, С) of the canonical fibre ^ = & XSxd (U X D) of the
mapping F.
Let -x
A.2.3) j*r = Я ®c,xa ^xo = *"F*C./- ® &S-XD
be the sheaf of holomorphic sections of the fibration Я, and V be the
integrable connection on Ж defined by the local system Я, i.e. the Gauss-
Manin connection. Obviously, the restriction of Ж to S' X {y},
Ж <8> (9's'x{y} = -^(y), is the sheaf of holomorphic sections of the fibra-
fibration H{y). By the relative Poincare Lemma we have a resolution
0 —> F~1(9's'xd —* Q>ys'x№ and the sheaf Ж is identified with the
sheaf of relative De Rham cohomology (chapter I, §3)
?' X D) = R-Fi(Q>/s,XZ)) = ^"(Fi(Q;
1.3 Canonical extension of the sheaf Ж and the meromorphic
connection
Recall ((II.6.3.5), (II.6.4.5)) that in the case of an individual singularity
fix) we constructed the canonical extension S§ of the sheaf Ж from S' to
the whole disk S by means of the trivialization of the sheaf Ж by
elementary sections s[A, a] = ^а(Л) = ta-tNA corresponding to multi-
multivalued horizontal sections A =A(t) € Hx, a = —(l/2yri)logA. We pro-
proceed in the same way for the sheaf Ж in the case of a family of
singularities.
Now let H be the space of multivalued horizontal sections of the sheaf Я
A.2.1) over S' X D. For an eigenvalue Я of the monodromy ? a section
A = A(t, y) € Hx and a value /J of logarithm —(l/2ra) log A, we define the
elementary section
A.3.1) s[A,p](t,y)=t^-tNA<<t,y).
This is a single-valued holomorphic section of the fibration H, i.e. s[A, /3]
1 Gluing of Milnor fibrations and meromorphic connections 143
is a section of the sheaf Ж. Since A is a horizontal section, it is clear
A1.6.4.2) that s[A, /3] satisfies the conditions
A-3.2) (tdt-p)H+x-s[A,P\ = Q,
dys[A, P] = 0,j=l,...,m,
where d, = Vd/d,, dj = Vd/dyj.
A.3.3) Definition Choose bases in the spaces Hx and let A\, ..., Ац be
the corresponding basis in the vector space Я = фНх. If we choose the
values cij of logarithms a = -(l/2;ri)logA of the eigenvalues correspond-
corresponding to the vectors Aj, we obtain sections, s[Aj, aj\ defining a trivialization
of the sheaf Ж over S' X D. If a.j are chosen such that condition
— 1 < a.j «s 0 holds, then the extension of the sheaf Ж corresponding to
this trivialization is called the canonical extension 3$ of the sheaf Ж to
SXD,
® 9{A ] 1< < 0.
Let
A.3.4) ^S
be the localization of the sheaf S along the subvariety {0} X D с S X D
defined by the equation / = 0. It is obvious that the restrictions of SS and
Ж to S X {y},
are the canonical lattice and the meromorphic Gauss-Manin connection of
the singularity f/. (C+I, 0) -> (C, 0).
For any of the logarithms /? = — (l/2;ri)logA of an eigenvalue of the
monodromy we denote by Cp the constant sheaf on S X D generated by
sections s[A, /?], A e Hx, and let Cp be the sheaf of holomorphic sections
of the fibration C)s|{o}xr>; Cp is a free ^o-module of rank fix = dim Hx-
We can also consider the sheaf Cp as the sheaf of germs of sections of the
sheaf Ж of the form s[A(t, y), /3] = t^-tNA{t, y), where A(t, y) depends
holomorphically on у and for a fixed у A{t, y) is a horizontal section of
my)-
Any section of the sheaf Ж can be written in the form
A.3.5) s = Y^gj(t,y)s{A3, aj],
where gfit, y) are meromorphic functions with poles along {0} X D.
Decomposing gj(t, y) into series in /, gj(t, y) = YlkgjA*' У)*к> and

144 /// The period map of a fi-const deformation
taking into account that tks[A, a] = s[A, a + к], we get a decomposition
of the form
A.3.6) s--
Thus, we have a decomposition
A.3.7)
where /? runs the arithmetic progressions —(l/2m)logA. The components
sp in A.3.6) are the (holomorphic) sections of the sheaves Cp. The
restriction of A.3.7) onto S X {y} gives the decomposition (II.6.2.5)
Ж(у) = ФрСр(у) of the meromorphic connection of the singularity fy
into the sum of root subspaces of the operator td,. As in the case of
individual singularities fy we often do not distinguish between ^M and
the restriction ^M\{q)x.d- There is a filtration V' (II.6.3.2) in Ж, and
2 Differentiation of geometric sections and their root components wrt
a parameter
When studying individual singularities in A.5.1.5) we associated with each
(и + l)-form со the section s\ui\ of the sheaf Ж, the geometric section of
Зв. The geometric sections generate the Brieskorn lattice .Ж@) С Л&.
Now we want to extend this to families of singularities and to learn to
differentiate the obtained geometric sections wrt the parameter.
2.1 Geometric sections and their root components
Let fy be a /г-const deformation of a singularity /: (C"+1, 0) —> (C, 0) as
in §1 and let w ? T(S\ Q"?)SXD) be a relative (и + l)-form represented
by the differential form
B.1.1) (o = g(x,y)Axof\...l\uxn,
where g(x, y) is a holomorphic function.
As in A.5.1.5) for a fixed y? D the restriction coy of the form со to
J&(y) defines a section s[et)y] of the fibration H{y) over S' X {y}: the form
(cy/d/y)@ = Resj-^Xco/ify — i) defines the cohomology class s[(oy](t)
in the fibre J?{y)t = &(uyy Varying у we obtain a section ,s[cy] of the
fibration H:
2 Differentiation of geometric sections
145
B.1.2)
s[co](t, y) =
CO
Wy
€ Hn{jS\tty), C).
In the same way as in A.4.2.7) one can verify that this is a holomorphic
section of the fibration H, i.e. s[cy] is a section 3@. We call s[co] the
geometric section of the form со.
We now consider the root components. By (II.6.5.6) for a fixed у € D,
s[ot)y] € SZ{y) and the root decomposition (II.6.2.6) for s[(oy] is written in
the form
B.1.3)
where the root components s(co, p) e Cp(y). By varying у we obtain a
section s(a>, /?) of the fibration Cp on D.
B.1.4) Remark s[eo] and ^(cy, /?) can be considered in two senses. On
the one hand s[(o](y) as a function of у only is a section of
•^\{o}-x.d = @pCp. On the other hand, these sections are considered as
germs of sections of the sheaf.Ж = ^\S'xd> S' X D = S X D\{0} X D,
i.e. s[co], as a function in t and y, is a section of M of the form
', y) =
B.1.5) Finally, we note that the sections s(cy, /S) of the fibrations Cp are
holomorphic, i.e. the s(a>, p) are sections of the sheaf Cp. Indeed, if we
present s[cu] as a section of the sheaf 3% in the form A.3.5), then the
functions gj(t, y) are holomorphic functions on S' X D because s[cy] are
sections of Ж and s[Aj, aj] is a basis of sections of this sheaf. On the
other hand, the functions gj(t, y) are holomorphic on the whole of
S X {y} for every fixed у because s[Aj, aj] is a basis of S§{y) and
s[(oy] e M@)(y) С Щу). Now by Hartog's theorem it follows that the
functions gj(t, y) are holomorphic on the whole of S X D. Decomposing
gj(t, y) into series in t as in A.3.6) we obtain that s(a>, p) are holomorphic
functions.
We now want to learn to differentiate s[ou](y) and s(oo, fi)(y) wrt the
parameter y.

146 /// The period map of a fi-const deformation
2.2 Formulae for derivatives of geometric sections and their root
components wrt a parameter
To differentiate s[co](y) along the parameter yj we have to consider s[w] as
a section s[co](t, y) of the sheaf J&, take its covariance derivative
dyj = Чэ/dyj and then again consider dy.(s[co](t, y) as a section of
Consider the following function on S' X D
B.2.1) 1шА*'У) =
o(t,y)
\-?V,y)
where a(t, y) is a flat section of the homological fibration. Since
dyjo(t, y) = 0, it is sufficient by A.2.7.3), in the same way as for calculat-
calculating d, = Vd/d/ in (I.4.3.I), to find
B.2.2)
/, y) = (dyjs[co](t, y), a(t, y)).
In the same way as in A.4.3.1), to calculate the derivative we can use
Leray's residue theorem A.4.2.6) to present the function 1ш,о(и у),
2.2.3) f "¦ ' Г
,,,» fy(x) - t'
where d is the Leray coboundary and coy = dfyAT), t}\^-(ly) =
(a)y/dfy)\j[-{ly). We can assume that the cycle d(o(t, y)) is the same in a
neighborhood of the point (t, y) and we can apply the standard theorem of
analysis when differentiating an integral wrt a parameter.
The derivative of the integrand on the right-hand side of B.2.3) is
9 ( wy \ _ д ( w \ -.dw 1 dF w
dy~ \Mx) - t) ~dy~j[F(x,y)-t) ~dy~j"FZ~t~dy'j'WrW'
We can transform the second summand using that wy — dfy Л rj and
dF
From this
8F
{fy(x)-tl fy(x)-t
dm
?«ЛЛ,
_
dyj
•v
\fyix) ~ t) fy{x) - ! fy(x) - t \fy(x) - tj
Integrating along d(o(t, y)\ we obtain by B.2.1) and B.2.3):
2 Differentiation of geometric sections
{dyjs[w](t, y), a{t, у)) =(*[Я('> У)' о('>
147
It remains to note that by A.5.2.1) d,((dF/dyj)co) = d,(dfy Л
= d(dF/dyj)V), i.e. s[d((dF/dyj)V)] = dts[{dFldyj)wl giving
B.2.4) dyJs[co](t,y) = sl9w
РуА
\dF
Finally, we obtain:
B.2.5) Theorem If a function F(x, y) defines a //-const deformation of
the singularity /: (C+1, 0) -> (C, 0), у € D с Cm and w - g(x, y)dxo
Л ... Л dxn is a holomorphic (л + l)-form, then for the geometric section
€ Л& we have
B.2.6) dy.s[w](y) = s И (y)-d,s Щ- J (y).
For root components s(w, a){y) this gives
B.2.7) dyjs(w, a)(y) = s(^-,a\(y)- d,s(^w, a + \\y).
Formula B.2.7) is obtained by comparing root components in B.2.6) since
dt: Ca -> Ca-i.
B.2.8) Corollary If a form со — g(x, y)dxo Л ... Л йх„ does not depend
on y, then
B.2.9)
B.2.10)
dF
dys[w] = -d,s\—w\,
dyjs(co, a) =
jZ-a, a+l\
i.e. to find dyp we ought to multiply со by —dF/dyj, and then to apply dt.
Since the operators dy\, ...,dym and d, commute, we can iterate
formulae B.2.6) and B.2.7) to find higher derivatives, and use them for
decomposition into Taylor series. As an example, examined by Hertling
[He], we consider the following.

148 III The period map of a fi-const deformation
2.3 Decomposition of the root components of geometric sections into
Taylor series for upper diagonal deformations ofquasihomogeneous
singularities
B.3.1) Let /: (Cn+1, 0) -> (C, 0) be a quasihomogeneous singularity
denned by the polynomial f(xo, ¦¦¦ ,xn) of degree deg/ = 1 with weights
wtx, = w,-. In this case by proposition A.5.5.4) the section s[w] defined by
the monomial (n + l)-form
со = xk dx = Xq°- ... • x\" dxo Л ... Л dxn
is homogeneous, i.e. it contains only one root component
B.3.2) s[a>] = s(w, v{co)) e CK(u),
where
B.3.3)
= a(k) - 1 =
i=0
In this case the order function a defined by the roof decomposition
(II.6.3.1) coincides with the Newton order v A1.8.4.4), a(s[co\) = v(w).
For later use it is useful to reformulate B.3.2) in the following form:
B.3.4) Lemma If со = xk dx, and M = xl = x!Q°- ... x[" is a monomial of
weighted degree v(M) = wolo + ¦ • • + wnln =: b(M) + 1, then s[Mco] =
s(Mco, v(Mco)), where
v(Mco) = v(M) + visa) = b(M) + 1 + v(w).
B.3.5) We now consider a //-const deformation of the singularity / Let
Mj = xlJ = Xqj- ... xnnJ, j = 1, ..., m, be monomials of weighted degree
v(My)>l.Then
bj = b(Mj) = v(Mj) - 1 > 0.
Consider the m-parametric deformation
f t f
B.3.6)
fy{x) = F{x, y) = /(*) +
of the singularity fo(x) = f(x\ which is, as is well known [AGV], а /л-
const deformation. Consider the geometric section s[co](y) induced by a
monomial form w = xk dx on D = Cm, and consider its root components
s(w, a)(y), which are sections of sheaves Ca on D. Let us find derivatives
of sections s(w, a)(y) and their decompositions into Taylor series at the
point у = 0.
Formula B.2.10) applied to the deformation B.3.6) gives
2 Differentiation of geometric sections
dyjs(co, a)(y) = -dts(x'Jco, a + lH>),
and iterating we obtain
B.3.7) dys(co, aXy) = (-5,)l'U(M'w, a + \i\)(y),
where the standard notation is used for
i = (/i i») € NJ1, |/| = и + ... + /m, i! = ii! ... /„I,
149
The decomposition of s(w, a) into Taylor series at the point у = 0 is of
the form
or applying B.3.7),
B.3.8) s(a>, a)(y) = ^H,I'1^^ a + ф^
i€N0" '¦
By B.3.4) it follows that s[M>w]@) = sW'w, v(M''<w@))), where
v(M'(w@)) = |i| + biii + ... + bmim + visa), bj = b(Mj) > 0. Hence,
s(M'a>, a + |;|)@)^0onlyif
B.3.9) a = bih+...+ bmiM + v{w).
Finally, we obtain:
B.3.10) Proposition For the deformation B.3.6) of a quasihomogeneous
singularity and for a monomial form со = xk dx the root components of the
geometric section s[a>] have a decomposition into Taylor series of the form
B.3.1
s(co, a)(y) = ^(-d,)
where the summation is taken only over the multiindices i € N™ satisfying
B.3.9).
In fact the Taylor series B.3.11) contains only a finite number of non-
nonzero terms. Indeed, the set of i e N™ satisfying B.3.9), i.e. the set of
integer points of the hyperplane b\z\ + ... + bmzm = a — v{m) in the
positive 'octant' is finite because all bj > 0. In particular:
B.3.12): if a < v(w), then s(w, a)(y) = 0; and
B.3.13): if a = v(<w), then
s(w, v(a)))(y) = const = s[<w]@).

150
III The period map of a //.-const deformation
2.4 The sheaves
We have considered how the geometric sections vary under ^-const
deformation of a singularity. We recall (II.7.2.1) that for an individual
singularity fy the Hodge filtration of the MHS on Hn(X(y)oo, C) is
defined by means of Gr^v.^°\y). So now we consider how the Brieskorn
lattices J^@\y) and their subfactors GrPv.j%@)(y) generated by the main
parts of geometric sections vary under the variation of y.
B.4.1) The geometric section s[w] of the sheaf 3? С -уМщхО on D was
denned for a relative (л + l)-form со in B.1.1). It has the root decomposi-
decomposition B.1.3)
s[co] =
where s(w, /3) is the holomorphic section of the filbration Cp B.1.5). Let
С 5§ be the subsheaf generated by geometric sections. The fibre
Мф) <g> &sx{y] is the Brieskorn lattice of the singularity fr
B.4.2) Definition The order of 'a geometric section s[co] is
One should distinguish a(co) from the orders of geometric sections 5
at points у
B.4.3) a(s[w](y)) = min {p\s(w, $){y) ф 0}.
For a geometric section s[co] = s(co, a(co)) + ... the main part
s(co, а(ш)) € Са(ш) can vanish at some points. Thus for all points у е D
B.4.4) a(s[co](y)) 3= a(co),
and the equality holds apart from a closed subset in D.
Thus the orders a(s[w](y)) of geometric sections at points can jump on
closed subsets. Nevertheless the subspaces Gr^yM^°\y) С СР(у) gener-
generated by s(w, a(w)) behave well. Indeed, by Varchenko's theorem (II.8.9.11)
the spectral numbers na remain constant under /г-const deformations.
Hence by (II.8.1.4") the dimensions of spaces
B.4.5) dim Gr^M(\y) = na + na+l + ... + nfi = /f,
also remain constant. Thus we have:
B.4.6) Proposition The vector spaces GrevJif{0)(y) С С&(у) form a sub-
3 The period map 151
fibration in C^, and the sheaf of holomorphic sections of this is a locally
free subsheaf Gri^<°> с Ср.
B.4.7) Corollary. For any /3 there exist relative (n + l)-forms
(O\fi, ..., штф, m = тф) = dimGrv3@W(y) — f%, on some neighbor-
neighborhood Uyv of a point yo € D such that a(coyjs) = a(s[a)j, ff](y)) = /3 for all
у е Un, and the main parts s((Djtp, /3) form a basis of Gr^^@).
B.4.8) Corollary For any (n + l)-form cdq € Q?t+i 0 in the neighborhood
of a point yo € D there exists a relative (л + l)-form со such that
s[w](yo) = s[coQ] and a(s[co](yo)) = a(co).
Indeed, if a(s[coo]) = /3, then it is sufficient to take со generating the
class s[coq] in Gr^v J^@) at the point y$.
B.4.9) Remark The vector spaces GravM°\y) ~ {s(co, a)(y)\
a(s[co](yo)) = a} are generated by the main (leading) components of
geometric sections. One can consider larger vector spaces {s(co, a)(y)}
generated by components of order a (not necessarily leading order) of
all geometric sections. The dimensions of these spaces unlike
dim Grav.W^\y) can jump. Their behavior is related to the Bernstein
polynomials.
3 The period map
Before considering the 'natural' period map denned by the MHS on a
vanishing cohomology we will first consider the period map defined by M.
Saito [Sa7] via embeddings of Brieskorn lattices 3@®\y) С Л> because
such an embedding defines the corresponding MHS.
3.1 Identification of meromorphic connections in a /г-const family of
singularities
Let F = FY: (C+1, 0) X Y -> (C, 0) X Y be a m-parametric /г-const
deformation of a singularity /: (C"+1, 0) -> (C, 0) with base Y,
dim Y = m, not necessarily local, / = /Л, y0 € Y. Let
C.1.1) F=FY: J&V-^SX У
be the corresponding 'Milnor fibration' arranged locally as in A.1.3).
If the base У is simply connected, then the elementary sections A.1.3)
s[A, /3] = t»tNA(t, y),

152
/// The period map of a ft-const deformation
where A(t, y) € Яд are multivalued horizontal sections of the fibration
H, make it possible to identify all meromorphic connections
yM{y) — (BaCa(y), у € Y, and all their homogeneous components Ca(y).
The sections s[A, P\ are horizontal along y, (d/dyj)-s[A, /3] — 0, and using
these we can form a basis of sections at every point y. Then the canonical
isomorphism ^M{y\) ^* Ж(уг) takes an element, which has some coordi-
coordinates relative to the 'elementary' basis, to the element with the same
coordinates relative to this basis. More explicitly, if A\, ..., Ацх is a basis
of Hi and s(y) is a section of the sheaf Cp, е~ъ"^ = X, then s(y) can be
written in the form
C.1.2) s(y) = ? gj(y)s[Aj, fi = J2 ^^ШЧ*, У)
and we can consider s(j) as a family of elements of the space CP(yo) with
coordinates gj(y) relative to the basis s[Aj, /3](yo). Thus we have the
canonical isomorphisms .Ж(у\) —* -J&iy?) for any two points y\ and
У2 € Y, which are isomorphisms of C{/}[r"']-modules commuting with
dt (or isomorphisms of C[<9(]-modules). Under the isomorphisms the V'-
filtration goes to the V- -filtration and, in particular, 2S(y{)—*Sg(yi).
Further we shall assume that a point yo is chosen, and we'll often write
*Ж{уо), 3?(y<y), etc., simply as ^#, 3>, etc.
If the base Y is not simply connected, then for any у € Щ(У, yo) we
have the automorphism hY: ,Ж(уо) —> ^Ж{уо) defined by extending sec-
sections along y. In this case to identify the connections .Ж{у) we have either
to consider ^S(jo) up to automorphisms hY, or to turn to the universal
covering Y -* Y (to consider the canonical fibre in the sense of §2,
chapter II).
3.2 The period map defined by the embedding ofBrieskorn lattices
Assume that we have a /г-const deformation C.1.1). Then we have the
Brieskorn lattice .%fm(y) at every point у е Y. Taking into account the
canonical identification ^S(y) -^ -Ж(уо) = Л6 we obtain a family of
Brieskorn lattices 3%®\y) С ^Ж in the meromorphic connection ^Ж
associated with the deformation C.1.1). By (II.8.3.7) we have
C.2.1) % = V>~1 D M\y) Э Vn~l
and the embedding Л?(°\у) С ^М is completely defined by the embedding
of finite-dimensional spaces
C.2.2) J^@)(y)/V~l С V>-l/V"-1 =: V.
3 The period map
153
Let us find the dimensions of these spaces. Since X]-i<aeo dim Ca = /г
and 2_1<а=гл-1 dim Ca = и/г, we have
dim V =
dim Ca = Щ —ц\,
where ^ = dim Cn = dim H'iX^, C)i.
From the symmetry of spectral numbers na = я„_1_а (П.8.3.3) it is easy
to obtain (one can see this visually in the picture on p. 107) the following
lemma:
C.2.4) Lemma
dim.%f@\y)/V-1 =
Denote by П' = Grass(k, V) the Grassman manifold of it-dimensional
subspaces in a vector space V, dim V = 2k= пц- }X\. Then П' is the
'space of periods' containing all Brieskorn lattices J^@)O)mod V"~x of
singularities fy /г-equivalent to the singularity / We obtain the period
map
C.2.5) Ф^-^П',^ M@\y)/V"-1
associated with the deformation FY of the singularity / defined by the
embedding of Brieskorn lattices. By B.4.6) Ф is a holomorphic map. In
general, Ф is multivalued if У is not simply connected.
C.2.6) The space П' is too large. We can construct a more 'economical'
space П с ГГ containing all Brieskorn lattices. Let Sp(f) = ?иа(а) be
the spectrum of the singularity/ Let
«min = min {a\na Ф 0}, araax = max {a\na Ф 0}
be the smallest and the largest spectral numbers, respectively. It follows
from (II.8.3.4) that amin and amax € (-1, ri), and from (II.8.3.3) that
Omin + «max = n — 1. In the same way as in (II.8.3.7) we can prove a
refinement.
C.2.7) Lemma
Э .
Э
Vn~l.
Proof By (II.8.1.4") we have GrPvM®\y) = 0 for ?<amin. Hence
m С Ka»». Furthermore, GrpvM0){y) = Cp for ?>amax-l

154 /// The period map of a fi-const deformation
because otherwise there exists a spectral number э= /3 + 1 > amax. This
yields М(й\у) D V>a™-\ ¦
C.2.8) It follows from C.2.7) that Ф(У) С П с П', where П is the
Grassman manifold of vector subspaces in Vam{n/ V> a'a"'1 of dimensions
C.2.9) To calculate the period map Ф: Y —* П explicitly, we have:
(i) To calculate the spectrum Sp(f) and to consider the vector
space V = Fa™»/F>a™'~1 = ®a,=eastf,Ca, where a\ = ат\а, and fa =
amax-1-
(ii) To take enough relative (n + 1)-forms coi(y), ..., a>\{y) so that
corresponding geometric sections s[a>\\ s[a>\] define the position of
the subspace JW(-0)(y)/V>^1 = (s[ct)i], ..., s[a>\]) in V. The problem now
is to learn how to find the vector functions s[a>j](y). Here we give two
methods which we'll use in what follows: A) if Fy is a semiquasihomo-
geneous deformation B.3.6), we can use proposition B.3.10) to decompose
s[co] into the Taylor series; B) one can find differential equations such that
the vector functions s[ct)j](y) are their solutions, and then solve them to
find these functions. If one succeeds in finding the vector functions
s[coj](y), then in order to write explicit formulae for the period map, the
Pliicker coordinates ? of a point in П, corresponding to the subspace
3e(V()(y)l V>$x, have to be expressed as functions of y.
Now we examine some examples in detail.
3.3 Example: the period map for E\2 singularities *
Let us consider a two-dimensional singularity /: (C3, 0) —> (C, 0) defined
by the function f(x, y, z) = x3 + y1 + z2. This is a quasihomogeneous
unimodular singularity of type ?12 in the classification of V 1. Arnol'd. The
normal form of singularities ?12 is ([AGV])
C.3.1)
axy
+z2.
This function defines a semiquasihomogeneous /г-const deformation of the
singularity / with base Y = С Э a. We want to calculate the period map Ф
associated with this deformation.
The singularity / is quasihomogeneous of degree 1 with variables of
weights w0 =^,wl= k, w2 =j. We have wt(xy5) = 22. By A1.8.4.8) the
forms
C.3.2) сокл =xkyla>o,co0
3 The period map
form a basis of the ^j,0-module .Ж@), and the numbers
155
form the spectrum of the singularity / If we order the spectral numbers
a,- = ki/42,1 =? i =? 12, the numerators &,- give the set
{-1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 43}.
Thus, amin = ai = -? («, = a(s[wom and amax = f2 and fa = amax
- 1 = j2- The spectrum Sp(f) is symmetric wrt the point (n - l)/2 = \.
From A1.7.2.6) we schematically visualise Ж and the inclusions
(Щ С Ж with the following picture
V
J_L
R »
+
Here J& = ©Ca, where a runs 12 progressions fa + n, a2 + n,
..., an + n, a\ + n + 1, « 6 Z, and the root subspaces Ca are depicted
as columns of height dim Ca = 1. Those parts of columns correspond-
corresponding to Gr^y.^0) с Са are hatched and those corresponding to
d-'Gr^\%^ = GrtyJlH-» с GrtyJgW are shaded.
In this case from C.2.8) the space Fis :^
C.3.3) V=Cai eC>,,diinF = 2,
and the space 3?(°)(a)/V>a™>-1 is one-dimensional. Therefore, the Grass-
Grassman manifold П = P1 is a projective line.
The embedding of a Brieskorn lattice J%m(a) С Ж is defined by the
geometric sections s[ctH](a),
i?f@)(a) = С-(ф>о, сцХа) + s(aH, fa)(a)) ф V>^
and the point Ф(а) е Р1 corresponds to the line
C.3.4) C-(s(co0, aiXe) + s(w0, fa)(a)) с Ca, e Сл = V.
To calculate s(a>o, a.])(a) and s(coo, fa)(a), we use their decompositions
into the Taylor series B.3.11). In this case m = 1 and monomial
M= Mi=xy5. By B.3.13) we have
C-3.5) s(oH, ai)(e) = ф>о](О),
because a(w0) = v(wu). To decompose s(w0, fa)(a), we have to find i from
B.3.9) for a = fa, 6, = v(M) - 1 = v(xy5) - 1 = i i.e. from the equa-
equation ^ = bi- i. This gives i = 1, and from B.3.11) we have
C.3.6) ф>0, fa)(a) = -dts[xy5(o0]@)a.

156 III The period map of a fi-const deformation
We take the vectors
b0 = s[wo](O) e Ca and bx = -d,s[xy5aH]@) e Cfit
as a basis of the space V = Cai © CPr The line C-(§obo + Z\b\) С К
corresponds to the point (§0: ?1) G P1. It then follows from B.3.4), C.3.5),
and C.3.6) that the period map is given by the formula
C.3.7) Ф: 7 = C->Pl =П, он (ha).
If По: §o Ф 0 is the open subset По = С С P1 with coordinate ? = ?i/?o,
then Ф: Y —> По С П is simply the identical map, % = a.
C.3.8) 77ге general case The singularity ?12 is one of the 14 exceptional
unimodal singularities of V I. Arnol'd. Normal forms of these singularities
are of the form [AGV] fo(x, y, z) + a-M, where fo(x, y, z) is a quasi-
homogeneous monomial, and M is a monomial of weight wt (M) > 1. The
spectra of these singularities can be found in Table (II.8.8.1). For all these
singularities we have: all dim Ca = 1, amin = a.\ <0, amax = аи > 1 and
a,- >/?i = а„ — 1 for j = 2, ..., fi. The remaining calculations and the
result are the same as for the singularity ?12. These calculations are carried
outin[Hel].
3.4 The period map for hyperbolic singularities Tp^r
The unimodal (two-dimensional) singularities of the series Tp^r have the
following normal form [AGV]
C.4.1) fa=xp + yq +zr + axyz, афО,
where
C.4.2)
111,
p =:_ + - + -< 1.
p q r
This is a ^г-const deformation with the one-dimensional base Y = С
= C\{0} Э a. Let us find the corresponding period map (see [Hel ]).
The singularities fa are not quasihomogeneous, but they are non-
degenerate with respect to the Newton boundary., For them fi = p
+ q + r — 1, and the monomials
C.4.3) 1, xyz, xk@ <k<p), /@ < к< q), zk@ <k<r),
form a basis of the Jacobian algebra Qfa, and the forms
C.4.4) MiCJo, (o0 = dx Ady Adz, 1 « i « ft,
where M{ are the monomials C.4.3), represent basis vectors of the space
Qf = J&0\a)/Jg<-l)(a). From (II.8.5.8) we obtain that
3 The period map
C.4.5) a([wo](a)) = 0, a(s[xyzco0](a)) = 1,
a([xkco0](a)) = у a([ykco0](a)) = -, a([zka>0](a)) =\,
and the spectral numbers are
C.4.6) Sp(/a) =
157
It follows from this that dim Co = dim Н"(ХЖ, C)^=! = 2. A diagram
based on (II.7.2.6) is of the form (the example shown is for the singularity
Jhh
-1
0 'Л'/з
For the singularity fa we have a, = amin = 0, amax = 1, ^ = am
-1=0 = a,. Therefore, V = Ka'/F>^ = C0)
a) D V
>0
, 0)(a) ® V
>0
C.4.7) V°
and the embedding .Ж<0)(а) с Ж is defined by the embedding of the one-
dimensional space C-s(w0, 0)(a) into the two-dimensional space V = Co.
To find s(co0, 0)(a), we will obtain a differential equation which has this
vector function as its solution. Since dfa/da = xyz, we have for
a>o = dx A dy A dz from B.2.8) that
C.4.8) das[w0](a) = -dts[xyzaH](a),
We now have to find a relation between the derivatives of the function
s(co0,0). Actually we already have a differential equation: since
s(co0, 0) e Co, Co is a root sub space of the operator td, and dim Co = 2,
we have
C-4.9) (tdtfs(u)o, 0)(a) = 0.
It now remains to express the operator
C.4.10) (tdlJ = d2tt2-3d,t+l
as da and d\ by means of C.4.8) and to use that the function fa = tin Ж.
To do this we employ an obvious identity which follows from C.4.1) and
C.4.2),
C.4.1
fa =
a(\ - p)xyz.

158 /// The period map of a [i-const deformation
To express dtts[wo](a) we multiply D.4.11) by co0 and apply the operator
d, to the corresponding geometric sections
C.4.12)
d,ts[wo]{a) = d,s[fa(o0](a) = d,s\ f-x/v + ... + a(l
We then have
C.4.13) dts[xf'a,xw0](a) = s[co0](a)
and analogous equalities for the derivatives wrt у and z. This follows from
the identity xf'xa)o = d/ Л т], where t] = x dy A dz, and from the rule of
calculation of the Gauss-Manin connection A.5.2.2), dts[xf'xco0] = s[dt}],
dt] = dx Л dy A dz = <y0-
It follows from C.4.12) and C.4.13) that
C.4.14) dtts[co0](a) = ps[a>0](a) + a{\ -p)dts[xyzw0]{a).
To express 6\ t2 s[wo](a), we multiply C.4.11) by faco0 and apply the
operator d,,
C.4.15)
dtt2s[w0](a) = d,s[f2aco0](a) }
...+ -rZfaf'a,z + fl(l -
(«)•
fJaj, r
The terms on the right-hand side (except the last one) are transformed by
means of the formula
C.4.16) d,s[xf'ajJa(Oo](.d) = s[faco0 + xf'ajOdKa)
= (t + d;l)s[w0](a),
which follows from the formulae xfaf'atXWo = dfa A (fax dy Л dz), and
d(fax dy Л dz) = faWo + xf'ajWo. We substitute fa from C.4.11) in the
last term on the right-hand side of C.4.15) and apply the formula
C.4.17) dls[x(xyz)f'a,xw0](a) = 2s[xyzwo](a),
which follows from the formulae x(xyz)f'atXco0 = d/ Л (x2yzdy Л dz), and
d(x2yzdy Adz) — 2xyzco0. Applying C.4.16) and C.4.147), we obtain from
C.4.15) that
dtt2s[co0](a) = p(t + a-'M[wo](a) + a{\ - p)BPs[xyz(o0](a)
+ a(\-p)-dts[(xyzJco0](ay).
Applying d, to both sides of the equality, we obtain
C.4.18)
3 The period map 159
d2t2s[co0](a) = pd,ts[wo](a) + ps[wo](a) + 2a(l - p)pd,s[xyzco0](a)
+ a2(l-pJd2s[(xyzJu>ol(a).
Substituting C.4.14) and C.4.18) into C.4.10), we obtain
C.4.19) с^2ф>0](а)
= A - p2){a2d2s[(xyzJco0](a) - 3adts[xyzw0](a) + s[co0](a)}.
Finally taking C.4.8) into account, we rewrite the differential equation
C.4.9) for the root component s(co0, 0)(a) in the following form
C.4.20) a2d^(a>0, 0)(a) + 3adas(aj0, 0)(a) + s(w0, 0)(a) = 0
or
C.4.21) {ada + V2s((o0, 0)(a) = 0.
The differential equation C.4.21) has the general solution cta~l +
c2a~l In a. Taking into account the initial conditions satisfied by
s(co0, 0)(a) for a = 1, we obtain that
C.4.22) s(w0, 0)(a) = bxa~x + b2a~x he,
where b\ — s(w0, 0)(l), b2 = b\ + das(co0, 0)(l), are vectors in the space
To summarize: the period space for the singularities Tp,q,r is the
projective line P1 of lines in Co. Take vectors b\ and bi as the basis in Co,
and let (|o." ?i) be coordinates of the line ?ob\ +%\b2. Then the period
map
Ф: Y = C* -> P1 = П
takes
a t-+ (a~l: a~l lna) = A: lna).
The map Ф: C* —> С С P1 is multivalued since C* is not simply con-
connected. If Y = С —> С* = Y, a »-> a = e°, is the universal covering of Y,
then the map Ф: У —» С С Р1 is the identity map, ? = a.
3.5 The period map for simply-elliptic singularities
The simply-elliptic singularities can be characterized from the point of
view of their desingularization in the following way. They are normal two-
dimensional singularities such that the exceptional curve of their minimal
resolution is a smooth elliptic curve E. The hypersurface singularities
among them are the singularities Ё$, Ё7, Ё6. These are those for which
(E2) - — 1, -2 and -3 respectively. In Arnol'd's classification of critical

160
/// The period map of a ц-const deformation
points of functions they are the unimodal singularities Jio = ^2,3,6.
X9 = Г2>4,4, Ps = 7з,зK respectively.
The Л-equivalent (or J2f-equivalent) classes of these singularities corre-
correspond exactly to the isomorphic classes of elliptic curves E defined by the
y'-invariant. Each elliptic curve can be written in the Legendre normal form
C.5.1) z2 = y(y - l)(y -X),Xe C\{0, 1},
and its y-invariant is j = j(X) = ±(X2 -X 4- \f/X2{X - IJ. Each of the
singularities Eg, Elt E(, has the corresponding Legendre normal form /я,
where Я € Y = C\{0, 1}. However, to calculate the period map it is
convenient to use another normal form. We'll review in detail the case of
the singularity E-i for which the calculations are the most simple. The
results for the singularities E(, and Eg are completely analogous [Hel].
C.5.2) The simply-elliptic two-dimensional singularities of type E-] have
the Legendre normal form
yx(y - x)(y - Xx) - z2, A € C\{0, 1},
and the corresponding stable equivalent one-dimensional singularities
gx = yx(y - x)(y - Xx).
For the construction of the period map we'll use another normal form:
C.5.3) /,(*; y) = x4 + ax2y2 + /, a 6 C\{-2, 2}.
The singularities /„ and gx are Л-equivalent for a = AX — 2. Form C.5.3)
is more convenient for calculations because the parameter a appears only
in one monomial.
Let us calculate the period map for the /г-const deformation C.5.3) with
the one-dimensional base Y = C\{-2, 2}.
The singularities fa(x, y) are quasihomogeneous with weights wo =
w\ = \, the Milnor number ц is equal to 9, a C-basis of ^@)(a)/
J&(~l)(a) is given by the forms
C.5.4) хку'ш0, where a>0 = dx A dy, 0 =s k, I =? 2,
and the spectral numbers coincide with the orders a(s[xkу1 сой](а)) of the
geometric sections of these forms, and by (II.8.4.8) these orders coincide
with the Newton orders v(xky'(o0). Hence,
C.5.5) Sp(fa) = {%k+ D + K/+ 1)- l|0 < k, /< 2}
— \ 2> 4' 4> u> U> ' 4' 4' 2J
A diagram based on (II.7.2.6) is of the following form:
3 The period map
161
-1 -У,-Уг -V*
0
Уг 3Л 1
ai=amin = -j, Pi=anax-l = -$ = a1. Hence, V=
C_i/2, dimC_!/2 = 2, and
C.5.6) У1'2 э .Ж@\а) D V>-x>2, M\a) = C-s[wo](a) © V>~xl2.
Note that the geometric sections s[co0](a) are homogeneous s[aH](a)
= s((D0, -^)(a). It follows from Taylor's formula C.3.11), s[wo](y) =
Т,&ъ(-д,Уз[х2у2У(о0]@)(у1/^, because all (a,)'s[(*V)'e>o](O)
6C-1/2.
Thus the embedding Ж(Щ(а) СЖ\% defined by the disposition of the
line C-s[co0](a) С C_1/2 = V. The period space П = P1 and the period
map is
C.5.7) Ф: Y = C\{-2, 2} -> P1, a ~ [C-s[<o0](a)].
To determine s[co0](a) we have to find a differential equation which has
this vector function as its solution. Since dfa/da — x2y2, we have from
B.2.8) that
C-5.8) das[co0](a) = -d,s[x2y2(o0](a),
and
%s[a)o](d) = d]s[{x2y2Jw0]{a).
We want to find relations between these derivatives. To do this, we express
(x2y2JaH in x2y2(o0 and co0 in :M/{0\a). First, we can write x2y2 from the
equation /„ = / C.5.3), and (x2y2J from equation C.5.3) squared
C.5.9) f\ = (x4 + y4J + 2ax2y\x4 + /) + a\x2y2J.
Second, we use relations which hold for the Brieskorn lattice
.%?(°\a) = Q2x/df A d&x A-5.3.3) in the case of one-dimensional singu-
singularities (n = 1): for any function g € C{x, y} we have d/o Л dg
= 0modd/a Л d&x, i.e.
C.5.10) (faXgy -fa,y-gx)w0 = 0 in :^°\a).
Applying this for g = x5y, xy5, we obtain relations in .
C.5.11) (х8-5Л>о = О,
Subtracting the second relation from the first, we get

162
or
/// The period map of a (л-const deformation
% + / - 10*V - 2ax2y2(xA + /))a»0 = 0,
= О.
1 - 12л:4/ - 2ох2/(л-4 + .
From this and from C.5.9) we obtain
/>0 = (A2 + aV/ + 4ал-У (л-4 + /))a>0 = 0.
Substituting x4 + y4 = fa - ax2y2 from C.5.3) gives
/>o - (A2 - 3aV/ + 4а/ал-2>-2)шо = О,
or
Now from C.5.8) and C.5.12) we obtain
C.5.13)
1
4a
Since the singularities ?7 are quasihomogeneous, the monodromy Г is
semisimple. Hence the root spaces Ca are eigensubspaces of the operator
td, with eigenvalues a. From the commutator relation [dt, t] = 1
d,t=td, +1, it follows that the operator djt2 - d,(d,t)t =
d,(td, + l)t = (<9//J + 3,/ is equal to (-iJ +± = f on the space C_1/2,
and the operator d2/ = dt(tdt+ 1) is equal to jdt on the space C1/2.
Finally, since the geometric sections s[co0](a) — s(co0, -j)(a) € C_\/2 and
¦у[*Ушо](а) € С\/2 are homogeneous, relation C.5.13) is rewritten in the
form
1 4a
dls[coQ](a) = u2H]()
or, from C.5.8), in the form
C.5.14) D - a2)d2as[co0](a) = \s[co0](a) + 2adas[co0](a).
Returning to the Legendre parameter Я = \ + \a (a = 41 — 2, да = \дх,
4 - a2 = -16A(A - 1)), we obtain that the function
is a solution of the differential equation
C.5.15) X(X — 1)дхи + Bл -
This is the hypergeometric differential equation with а = @ = \, у = 1.
C.5.16) Now let Аи А2 be a Z-basis in Я'^оо, С)-ь and let A{(t, a),
A2(t, a) be the corresponding multivalued horizontal sections of the
3 The period map
163
fibration H-\. Then the elementary sections e\ = s\A\, — j](a),
e2 = s[A2,-±](a) from A.3.1) form a basis in C_i/2. Now if
s[co0](a) = I\(a)e\ + I2(a)e2 is a representation of s[wo](a) in this basis,
where I\(a), I2(a) are (multivalued) holomorphic functions on
Y = C\{-2, 2}, then the period map C.5.7) is given by the formula
Ф(а) = G,(а): /2(а)).
Since the basis e\, e2 is horizontal, dae\ = dae2 = 0, and s[wQ](a) is a
solution of C.5.15), we have that /,(a) = /,DA - 2), I2{a) = I2DX - 2)
are also solutions of the hypergeometric equation C.5.15). Thus, the period
map Ф is given by hypergeometric functions. The classical theory of such
functions enables us to obtain more detailed information about Ф.
C.5.17) The same results occur for singularities of types ?8 and E6.
Analogous but more tedious calculations were done by Herding [Hel] who
showed that as in the case of E-, the function s[co0](X) satisfies the
hypergeometric equation C.5.15). We refer the reader to [Hel] for the
details.
It is useful to compare the period map Ф, obtained above, with the
period map of the modular family of elliptic curves, described in detail in
§9.3of[Br-K].
C.5.18) Hertling [Hel] has also calculated the period map for the bimodal
singularity WXfi (and other bimodal singularities in [He2]), which has the
normal form
Л*(*, У) = х4+у6 + ах2уъ + Ъх2у\ аф2, -2.
This is a two-parametric /г-const deformation with the base
Y = (C\{-2, 2}) X C. Herding showed that in this case the 'true' period
space П is a two-dimensional manifold which is a Stein fibration over Pl
with a Chera class equal to 2. The calculation of the period map leads to
the hypergeometric equation with a = ^, E = j, у = ~.
3.6 The period map defined by MHS on the vanishing cohomology
A natural development of the ideas related to the period map of a family of
non-singular projective manifolds leads to the period map which in the
case of the ^-const deformation A.1.3) F: .?' —* S X Y of isolated
singularities fy = F(., y) sends a point у б Y to the mixed Hodge
structure on Н'^Хуоо), Z),
у >-> MHS on Hn(^\y)x, Z).

164
III The period map of a fi-const deformation
However, we began with the period map defined by embedding of
Brieskorn lattices because the MHS is defined by the embedding
.Jf?{0)(y) С .Л and the V--filtration on ..M by virtue of A1.7.2.1). The
weight filtration of the MHS is defined by the monodromy (i.e. by
topology) and it is unchanged under /г-const deformations. We haven't
considered yet the question of the polarization of the MHS on the
vanishing cohomology [S3, V10, Ka2]. Disregarding the question of the
variation of the MHS (see the review [B-Z]), we mean the period map here
to be a map which sends a point у to the Hodge filtration F'(y) of the
MHS on #я(лГООос, 2).
C.6.1) Denning the period space П' = {.^<°>(>;)|V>-1 D M°\y) Э
Vn~1} = Grass(k, V) in C.2.5) and C.2.8) we viewed M0)(y) only as a
vector space in Ж. However, Ж@)(y) is a C{ ?}-module invariant wrt the
operator d~x acting on % = V>~x (Жт(у) is a
f'
}}-тоAи1е),
and there is a V~ -filtration on .Ж. Let П be the subset of vector subspaces
W in J6, V>-1 D WD V"-\ i.e. W € ГГ, such that If is a
C{ t} { {<9~'} }-module, and such that the dimensions
C.6.2) ma = dim Grav W, oeQ,
are given.
Recall A1.7.2.1.) that the Hodge filtration F'(y) on Нп(Ж(у)х, С)
t- ,<?{y)lt.c?{y) = <В-1<ачоСа(у) is defined by СгуЖт(у). Namely, by
A1.7.2.3) we have
and
C.6.3)
= dim FpCa = ma+n
C.6.4) Let П be the manifold of flags F' for which the dimensions
f = dimF'' are denned by equalities C.6.3) as mp. Here П is the period
space for Hodge filiations. The rule A1.7.2.1) defines the flag F'{W) in
terms of W e П, and therefore defines a holomorphic map П —¦ П.
C.6.5) Definition Let fy, у € Y, be a //-const deformation of an isolated
singularity. The map
where F(y) is the Hodge filtration on H"{.?\y)oo, Z), is called the period
map defined by MHS or, for short, the MHS-period map.
4 The infinitesimal Torelli theorem
We have a commutative diagram
Y Д П
165
П
The period map Ф, is a holomorphic map by B.4.6) in the same way of
Ф is.
C.6.7) The Hodge filtration F(y) is a rougher invariant than Ж@){у).
In going from the embedding of the Brieskorn lattice .Ж40)(у) с .Ж to
the Hodge filtration F'(y), i.e. in going from Л'т(у) to
GrvM(K){y) с Grv. M, we lose information. As an example let us look at
the period map Ф for the /г-const deformation C.3.1) of the singularity
.Ei2. For E\2 all spaces Ca are one-dimensional, dimCa(y)= 1. Hence
FpCa{y) is either 0 or the whole space Ca{y). Thus Fp(y) does not vary
and ФG) is one point, and at the same time Ф: Y = С —> П is an
embedding by C.3.7).
This example is a particular case of the behavior of period maps for /л-
const deformations of quasihomogeneous singularities which we'll con-
consider in the next section.
C.6.8) Remark The information lost (when we go from .3$?(°\y) to
Grv.%?@\y)) is expressed by M. Saito in terms of some linear maps cpa in
[Sa7].
4 The infinitesimal Torelli theorem
4.1 The V-filtration on Jacobian algebra. The necessary condition for
fi-const deformation
Let
D.1.1) fy(x) = FY(x, y): (C+\ 0) X (C", 0) -+ (C, 0)
be an m-parametric deformation of an isolated singularity fo(x) = f:
(C"+l, 0) —¦ (C, 0). There is a canonical map of the tangent space T0Y to
the base of deformation Y = Cm at the point 0 e Y to the Jacobian algebra
Qf — ?/c+\o/Jf (the Kodaira-Spencer map),
д 3FY
D.1.2)
ToY-+Qf, —
dyj
mod Jf.
The deformation D.1.1) is miniversal if this map is an isomorphism. The
miniversal deformation is usually obtained in the following way [AGV].

166 /// The period map of a fx-const deformation
We choose functions (monomials) M\(x)=\, M2(x), ..., M^x) €
<?C"-',o representing a basis of Qf, and consider the deformation
fx(x) = F(x, X): (C"+1, 0) X (C, 0) - (C, 0),
defined by the function
D.1.3) F(x, X) =f(x) +Ai + X2M2(x) + ... + ЛрМц(х).
The base Л = С of the miniversal deformation is identified with the
Jacobian algebra as in D.1.2)
D.1.4) 7оЛ ^ Qf, j-»Mj(x).
Any deformation D.1.1) is induced (modulo isomorphism) from the
miniversal one by a base change cp: Y —> Л, X — X(y), FY(x, y) =
F{x, X(y)). The map of the tangent spaces d<p@): T0Y -> ГОЛ,
4~d("@)G
Wdx,
is interpreted under identification D.1.4) as the map D.1.2)
д dFY
——i—>
dyj dyj
Now we introduce the V'-filtration on the Jacobian algebra Qf ~ 7оЛ.
There is a V- -filtration on the Brieskorn lattice .Ж/@) с .Ж induced from
the meromorphic connection .Ж (И.6.3.2). The filtration on .Ж<0) induces
the quotient filtration (II.8.4.5) on
The filtration on Qf defines a filtration on End Qf, where Qf is a Qf-
algebra. The multiplication by an element of Qf defines the endomorphism
of Qf. We define the V'-filtration on Qf, the filtration induced from
End Qf,
D.1.6) VйQf = {g€ Qf\gVpQf С VP+aQf, Щ.
Now let D.1.1) be a ,M-const deformation as in A.1.1).
D.1.7) Theorem [V-Ch] If fy(x) D.1.1) is a m-parametric ^-const defor-
deformation of a singularity /with base Y, then
d(p(T0Y)c VlT0A=VlQf,
i.e.
D.1.8)
j=
infinitesimal Torelli theorem
167
Proof. Let an element [&>o] € Й/- be represented by an (n + l)-form Wq-
Since .M{~^ = д~\Ж@)), we have by (II.8.4.5) to prove that for
idfy/dyj)\y-Q there exists an (n + l)-form rj such that
Let a) = co(y) be a relative (n + l)-form such that condition B.4.7),
s[(o]@) = s[w0] and a(s[(o](yo)) = a(w),
holds in a neighborhood of the point 0 € Y. Then
a(dyjs[a)](y) 2= a(i
Apply the operator djl to formula B.2.6),
da>
dys[a)](y) =
We obtain
Hence
Ш
= a(s[co0])
D.1.9) Corollary If f(x) is a quasihomogeneous isolated singularity, then
singularities fy{x) of its ^-const deformation are quasihomogeneous or
semiquasihomogeneous.
This follows from the fact that the ^nitrations on .Ж@), Qf and Qf
coincide with the Newton filtration defined by weights wq, ..., wn
(II.4.4.7), and Vх Qf = {g€ Qf\v(g) > 1}, and ГЦ) = 1.
4.2 Calculation of the tangent map of the period map. The horizontality
of the MHS-period map
Let fy(x) be an m-parametric ^-const deformation as in C.1.1), and Ф:
Y ~* П', П' = Grass(k, V), be the period map defined by the embedding
of Brieskorn lattices C.2.5).
D.2.1) As is well known, the tangent space to the Grassmann manifold ГГ
at a point [W] ? П' where W с V is a subspace of dimension dim W = k,
is identified with the space

168 /// The period map of a/i-const deformation
T[W]TL' =nom(W, V/W)
in the following way. If a vector ? e T[W]TL' is represented by a curve
W(t), W@) = fF, in IT, then the homomorphism §: W -+ V/W corre-
corresponding to ? takes a vector v e W to the vector ?(u) =
(d/dt)v(t)\i=0modW, where z>(/) is a curve in V such that v@)=v,
v(t) € fT(O.
Now let us calculate the tangent map
<№@): T0Y-> T[W]TL',
where W = .Жф\0) / V "~1 CV = V>~x/V,
D.2.2) Lemma If ? = Hcjd/dyj € Г0Г, then
4 The infinitesimal Torelli theorem
169
i.e. if an element of fF is represented by the geometric section s[w0],
coo 6 fic*+'.o>then
D.2.3)
d<E>@): s[co0] >-> -9,5
w0 •
Proof By D.2.1) to calculate dФ@)(?) we have to take v(y) e .Ж^\у),
and then to differentiate (cto/<9?)|r=0. Let us take the section v(y)
= s[ao](y) = Res [co0/(fy(x) - /)]. Then by B.2.9) we have
ds[a>0](y)
D.2.4) 77ie tangent map to the period map defined via MHS Now
let П С П' be the submanifold defined in C.6.1), П be the flag manifold,
and Ф: Y —> П be the MHS-period map from C.6.5). The tangent map
d4> is calculated in the same way as d4> in D.2.2). Since П С П?=1
Grassif", H), where H = H^XX, C) = F°, we have Г*П С ®J=1
Hom(F», H/F"), where * e П corresponds-to the flag F'@). The Hodge
filtration F' on H was defined in (II.7.2.1) by the filtration on
J?'jt5S = Ф-KasoCa and by the isomorphism CA/t lA-^ H, where
~ F?Ca =
с Са+Я.р,
~\<a =? 0.
Therefore, to calculate
D.2.5) d<F(O): Г0У-^Г*П,
we can take s[co0] € К"""-''.^*0*, the section s[co0](y) =
Res [шо/(/у(д:) - /)], and apply formula D.2.3) to it. By D.1.8) we have
s[(dfy/d%)\y=0(Oo] € Уа+"-Р+1Л/@)тоАУа+"-Р+х.Ж(-х\ and hence
AФ@) takes F* to Fp^/F" С H/F?.
We define the horizontal subfibration ThH of the tangent fibration 7TI
to the flag manifold П by the condition: at the point * € П corresponding
to a flag F" we have
Г'ПС ф Hom(Fp, Fp-l/Fp)= 0 Hom(Fp/Fp+\ Fp~l/Fp)
* +i +i
= ф Horn (Gr?,
+i
We have then obtained the following:
D.2.6) Proposition The MHS-period map Ф: У —> П of a ^t-const defor-
deformation/^ is horizontal, i.e. 6Ф takes values in ThTL.
This means that for the variation of MHS defined by a /г-const deforma-
deformation, Griffiths' transversality condition
С QlY <8> Cf
holds.
4.3 The infinitesimal Torelli theorem
Let Dft С Л be the /г-const stratum in the base of miniversal deformation
of a singularity / Then DM can be viewed as an analog of the moduli
spaces of projective varieties. Let Ф (or Ф) be the period map DM —> П.
Assertions about the injectivity of Ф, i.e. assertions stating that a singular-
singularity is uniquely determined by its periods, are called Torelli (type) theorems.
We saw in C.6.7) that in general Ф is not an embedding. We'll show that
Ф is an embedding locally at non-singular points of D^.
D.3.1) Theorem ([S7]) If fy is an m-parametric /г-const deformation
C.1.1) of an isolated singularity / = /o such that the images of
(<9/y/<9yi)|r=o, ¦••, (dfy/dy^y^o in Q/ are linear independent, i.e. by
D.1.5) dcp(O) is a monomorphism for the map <p: Y —> ГГ inducing fy
from the miniversal deformation, then the tangent map dФ@) of the period

170
/// The period map of a /г-const deformation
map Ф: Y —» ГГ defined by the embedding of Brieskorn lattices, is also a
monomorphism, and hence, Ф is locally an embedding.
Proof If ^ = J2cj(d/dyj)eT0Y, and § ^ 0, then d<p(O)(§) =
(dfy/d%)\y.o?O in Qf, i.e. (dfy/d?)]y=0? Jf. Consequently, for
w0 € .Ж@> the product @/,/0|)|,=oroo $- •^fM) С .^f@) or
/у/д?)|>=о<Уо] ^ •^(~" if •^f<0> is viewed as a submodule of Ж. But
-'> = 9-'J??@). Hence <9,s[(d/v/(9%=ocyo] ? .Ж@), i.e. this element
is not equal to zero in V/W= У>-1/.Ж^\0). Thus, <№@)(?) 7^ 0 by
D.2.3).
In particular, if D^ is the /г-const stratum in the base of miniversal
deformation,
D.3.2) D^CA, Dl = D
then Ф: ?>° —> П locally is an embedding.
D.3.3) In [Sa8] M. Saito defined the period map Ф: Y -» П for the case
of a /г-const deformation Fy: (C"+1, 0) X Y -* (C, 0) X Y with a not
necessarily non-singular base Y To avoid problems with the topological
triviality of /г-const deformations A.1.3)-A.1.5) M. Saito uses Deligne's
sheaves of vanishing cycles A1.8.11.8). Let X С C"+I be an open subset
containing 0, У be a reduced analytic space, and F — Fy: X X Y —> С be
a holomorphic function. Assume that fy — F\xx{y}' X x {у} ~* С nas an
isolated singularity, and Smgfy = {0}, fy@) = 0. Then F is a /<-const
deformation of an isolated singularity with base Y Instead of the fibration
H on (C\{0}) X Y in A.2.1) M. Saito considers Deligne's sheaf of
vanishing cycles q>fCxxs which is concentrated on Y = {0} X Y. This is
a constant sheaf, and its fibres at points ye У are a vanishing cohomology
of singularities fy. This enables us to identify the vanishing cohomology at
different points у e Y. Again using Deligne's sheaves M. Saito shows how
to glue the Gauss-Manin differential systems of singularities fy to a
locally constant sheaf on Y Operating in the category of D-modules he
proves theorem D.3.1), and its generalization to the case of an arbitrary
base У as a corollary of this theorem.
D.3.4) Theorem [Sa8] If D^ С A is the /<-const stratum in the base of
miniversal deformation of an isolated singularity /, then the period map Ф:
Dp —> П has zero-dimensional fibres, and by restricting to an open subset
in Dp, we can assume that the fibres of Ф are finite.
4 The infinitesimal Torelli theorem 171
4.4 The period map in the case of quasihomogeneous singularities
D.4.1) The /<-const stratum. Let f(x) be a quasihomogeneous singularity
of degree 1 with weights w0, ..., wn. Let M\(x) = 1, M2(x), ..., MM(x)
be a monomial basis in Qf and F(x, X) be the miniversal deformation
D.1.3). The quasihomogeneity of f(x) means that the morphism /:
(C"+1, 0) —> (C, 0), t = f{x), is equivariant with respect to the action of
С*Эг on Сй+1, то(х0, ..., х„) = (zw"x0, ...,тк"х„), and on C,
r(t) = zt. We can make the deformation D.1.3), F(x, X): C"+1 X С -> С,
equivariant also if we define weights of A7 such that wtA7A/7(x) = 1, i.e.
we put
D.4.2) Vj = wt A7 = 1 - wt M7(x),
and define the action of C* on the base С = Л by the rale
г о (A,, ...,Xn) = (tv>Xu...,Tv>'Xli).
We can decompose С = Л = Л+ХЛ°ХЛ" into a product according
to the weights of the C*-action. The deformations from Л0 are quasihomo-
quasihomogeneous and are defined by the diagonal monomials, and the deformation
from Л~ are defined by the upper diagonal monomials. By corollary
D.1.8) ([V6]) for the quasihomogeneous singularity the /<-const stratum
Dp = {0} X U° X Л~,
where U° is an open subset in Л0, is therefore non-singular.
Let us consider the period map Ф and the MHS-period map Ф for the
stratum Dp. Let л: U° X Л" -» U° be the projection, and j: U° =
U° X {0} -> U° X Л" be the closed embedding. We have the diagram
D-4.3)
Л_ П
\Ф |
U° -+ П
By theorem D.3.1) Ф is locally an embedding. In [S7] it is shown that the
Hodge filtration F- on H"{XX, C) is unchanged under upper diagonal
deformations, i.e. when we add upper diagonal monomials to a quasihomo-
quasihomogeneous singularity. Hence the period map Ф is constant on the fibres of
л, and consequently, Ф factors through ж. In particular, if the dimension
of a fibre is greater than 0, then the Torelli theorem is false for Ф. On
the other hand, if a singularity is quasihomogeneous, then on passing
_^@) h_> Gr'yJ&W we do not lose any information, and the maps Ф and Ф
are equivalent on U° <-*• D^. We thus obtain the following result of M.
Saito: J
D.4.4) Theorem [Sa8] If D^ is the /i-const stratum in the base of the

172
/// The period map of a fi-const deformation
miniversal deformation of a quasihomogeneous singularity, then there is a
projection n: DM —> U° to a closed submanifold U° С Dp, such that the
MHS-period map Ф: Dp, —» П is constant on fibres of n, and the map Ф is
locally an embedding on U°.
5 The Picard-Fuchs singularity and Hertling's invariants
In this section we'll be concerned with the 'global' Torelli problem. In §3
for a //-const deformation fy, у е Y, of isolated singularities we defined
the period map Ф: Y —> П to the Grassman manifold П, taking a point у
to the Brieskorn lattices .^@)mod V"~x in the vector space V>~1 /V"~l
(or in a more 'economical' space). If the base Y is not simply connected,
then the map Ф, in general, is multivalued. So before going into the
question of the injectivity of Ф, we ought to consider the analog of
Griffiths' modular space D/T, i.e. to take the quotient of П by an
appropriate group. There is another reason for this. In singularity theory
singularities are usually considered within equivalence relation. For exam-
example, the Л-equivalence is considered for function germs. This leads to the
equivalence relation '~' on Y. To have a map from Y/ ~ to П, again we
have to take a quotient of П by some group.
5.1 The Picard-Fuchs singularity PFS(f) according to Varchenko
E.1.1) The Picard-Fuchs equation and the period matrix Let /:
(C"+1, 0) —* (C, 0) be a germ of a holomorphic function with an isolated
singularity. With a misuse of language we will not differentiate between a
holomorphic (n 4- I)-form w and the geometric section s[co] determined by
it. Let cou ..., cOf, be a basis in the lattice .Ж/@) which we'll write as a line
Ш = (со i,..., (Dpi). Let F(t) be the matrix of connection coefficients of the
Gauss-Manin connection wrt the basis со. Then T(t) is the matrix of the
operator d, wrt the basis су,
д,Ш = a>T(t), д,ш} =
The system of linear differential equations
y1 = A(t)y, where A(t) = T\t),
is called by Varchenko [AGV] the Picard-Fuchs equation of the singu-
singularity /wrt the basis со A.8.1.5). This rystem has the following meaning. It
expresses in coordinates the condition of horizontality of a section
у = ш*у of the sheaf Ж* = #* <g> 6 $¦ wrt the dual connection V* wrt
5 The Picard-Fuchs singularity and Hertling s invariants
the basis To*
173
(со*
. со*) dual to the basis Ш. Here у = (ух ... y^y is
the column of coordinates of у wrt the basis Ш*.
If у — y(t) is a family of homology classes, i.e. a horizontal section of
the sheaf Ж*, then the integrals
y(/)
is a
give a solution of the Picard-Fuchs equation A.8.1.4.). If yx, ...,
basis of horizontal sections, then the period matrix
Q@ = Qj,@, where Щ. = [ ^ = (ш„ у}),
hi aJ
is the fundamental matrix of solutions of the Picard-Fuchs equation.
Let Я* be the vector space of solutions of the Picard-Fuchs equation.
H* can be regarded as the space of multivalued horizontal sections of the
homological fibration #*. The homology with Z-coefficients defines a
lattice #| с Я*. The lattice H\ is invariant wrt the monodromy M.
E.1.2) The Picard-Fuchs equation, i.e. the matrix A{t), depends on the
choice of the basis ы. The change to a new basis transforms this matrix as
follows. If cb = ((o\ ... <Ьц) is a new basis in .Ж@) and C(t) e
GL(ji, &s,o) is the transition matrix to this basis, cb = coC(t), then the
matrix Q{i) = (C'@)~' is the transition matrix to the basis &*,
о) = Ш*Q(t). If у is the column of new coordinates, у = со*у = со*у,
then the new coordinates are related to the old ones by
y=Qy-
The substitution у = Qy transforms the equation y' = Ay into the equa-
equation
У = A~y, where A = Qr'AQ-Q-'Q1.
E.1.3) Definition A system y' = A{i)y with a lattice Wj_ с W in the
space of solutions W of this system, invariant wrt the monodromy M, is
called the framed system. The lattice Wz is called the framing of the
system.
E.1.4) Definition Two framed systems {A{t), Wj) and 04@, Wj) are
called equivalent if there exists a substitution of coordinates у = Qy,
Q € GL(pi, 6>'s,o) which transforms the system y' = A{t)y to y' = A(t)y,
and the lattice Wz С W to Wz С W, Wz = QWZ. The singularity of a
framed system is the equivalence class of this system.

174 III The period map of a /u-const deformation
E.1.5) Definition The Picard-Fuchs singularity PFS(f) of a singularity /
is the singularity of the framed system (A(t), Wj), where A(t) = T'(t),
and F(t) is the matrix of connection coefficients wrt a basis
¦'* = Hic н*.
E.1.6) The Picard-Fuchs singularity and the period matrix. Choose a
basis у = (уь ..., у ft) in Wi and write coordinates of the basis vectors,
Yj 6 ^z to columns of a matrix Q. We can call the matrix Q the period
matrix of the framed system (A(t), W-?).
In fact to give a framed system is to give its period matrix. The matrix
A(t) and the lattice Wj_ are uniquely determined by Q(t). Indeed, since
Q(t) is the fundamental matrix of solutions, we have Q' = AQ and hence
A(t) = Q'(t)Q~l(t). The lattice Wz is the lattice generated by the columns
of the matrix Q{f).
The matrix Q depends on the choice of the bases w and y. The passage
to new bases w = WC, С e GL(ju, f- s,o), and у = yU, U 6 GL{fx, Z),
transforms the matrix Q to the matrix
(one can obtain this very quickly by writing Q as a product of matrices
Q = w о у, where со,- о yj = (со,-, yj)).
Thus, we can regard the singularity of a framed system (A(t), Wj) as the
equivalence class of its period matrix wrt the action of group
In [AGV] Varchenko proposed the following conjecture.
Varchenko's conjecture The Picard-Fuchs singularities of non-/?-equiva-
lent germs of functions with isolated singularities are different at least
locally, i.e. when the germs of functions lie sufficiently close to each other.
We wish to reformulate this conjecture in terms of the period map. For
this we need to introduce an equivalence relation on the set of Brieskom
lattices (to introduce an action of a group on the manifold П).
5.2 The Hertling invariant Herx (/)
There are two reasons for introducing an equivalence relation on the set of
Brieskom lattices. The first is connected with the period map of families of
singularities with non-simply-connected bases, and the second with the
construction of invariants of/^-equivalence classes of singularities.
5 The Picard-Fuchs singularity and Hertling's invariants
175
E.2.1) The horizontal monodromy. Let F = Fy: (C"+1, 0) X Y ->
(C, 0) X У be a ^-const deformation of an isolated singularity / = /л,
yo € Y. In the case of a simply connected base У we defined in C.2.5) and
C.6.1) the period map
Ф:Г-.П,^ .Жт(у) с.Л = .Щуо),
and also the MHS-period map
Ф: Y -> П, у н-> F-(y),
where F'[y) is a flag in the vector space H = Hn(X00(yo), С).
If У is not simply connected, then for any у &n\{Y, y0) there appears
the monodromy transformation p(y) of H, which we'll call (to distinguish
it from T) the horizontal monodromy corresponding to y. We obtain a
representation p of the group n\(Y, y0) on the vanishing cohomology Я of
the singularity /л. If Y is the base of the miniversal family, i.e. if
Y = Z)" с Л is the manifold of non-singular points of the ^-const stratum
in the base of this family, then we can call the group G/, = р(л\(О0)) the
group of horizontal monodromy of the singularity f.
The horizontal monodromy commutes with the monodromy T (this
follows from the group n\(Sl X S1) = Z2 being abelian) and, obviously,
preserves the lattice Hz С H. Hence Gh С Gi, where
E.2.2) G, = {g 6 GL(Hz)\gT = Tg}
is the subgroup of integral automorphisms of H commuting with the
monodromy T.
The group Gi naturally acts on the set of flags in H and, consequently,
acts on the manifold П. In order to be able to define the period map in the
case of a non-simply-connected base Y, we can consider the quotient
П/Gi, as is usually done in the theory of period maps A1.2.3.3).
We can proceed similarly in the case of the period map defined by
embedding of Brieskom lattices. For у 6 Jti(Y, yo) this is the monodromy
transformation p(y) of the space .Ж = . /6{yo). We can define the action of
the group G\ on .Ж and, consequently, on П. We'll return to this presently.
E.2.3) The invariants of ^-equivalence classes. The point Ф(у) б П or
Ф(у) 6 П in the period space is not an invariant of the /{-equivalence class
of a singularity fy. Let us consider how these invariants transform under a
change of coordinates.
Let /i and /2: (C"+1, 0) -> (C, 0) be two /^-equivalent singularities.
This means that there exists a holomorphic change of coordinates
g: (C"+1, 0) -> (C"+I, 0) such that/, =/2og, i.e. /,(*) = f2(g(x)). The
morphism g induces an isomorphism of the Milnor fibrations X\ —* S and

176
/// The period map of a fi-const deformation
X2 —» S which commutes with the monodromy T and preserves other
topological invariants (i.e. preserves the intersection form, the bilinear
Seifert form, transforms one distinguished basis to another distinguished
basis). Now if f\ and f2 belong to a ,u-const deformation, f\ = /V|,
f2=fy2, then under canonical isomorphisms Щу\) ~ H ~ H(y2)
(respectively, . ?6{y{) ~ . /6 ~. /6(уг)) the filtrations F\y{) and F\y2)
(respectively, the Brieskorn lattices .Ж<0)О>1) and ¦^{0\у2)), in general, do
not coincide, and again we see that there exists an automorphism g € G\
such that F-(y\) = g(F(y2)). Thus, under the map Ф,:7^ П/G, the
points Ф^уО and Ф\(уг) coincide, and Ф\(у) gives the real invariant of
/^-equivalence class of the singularity fy(x).
E.2.4) Now we consider the action of the group G\ on .Ж. Recall that we
have constructed (II.6.4.8) an embedding of the vanishing cohomology
vector space in the meromorphic connection
rp: Hn(Xx, C) = tf ^ Ф Ca С .
10
The automorphisms g € GL(H) are transferred by means of \p to
Ф-i <a^oCa, g *-> ^ ° g ° V~'> and *hen are extended to C{t}[rl]-auto-
morphisms of .Ж. We obtain an inclusion GL(H) —> Aut.J%, g 1—» ij>(g).
We showed (П.6.6.4) that g € GL(H) commutes with the monodromy Tif
and only if r/j(g) commutes with dt, i.e. ip(g) is a C[9,]-module auto-
automorphisms of.Ж. Let Г1 = ^(Gi) be the subgroup of Aut.^ correspond-
corresponding to G\ с GL(H). The group Г| preserves Ca and hence acts on the
vector space V = V>~1 /Vn~x and, consequently, acts on the Grassman
manifold П.
E.2.5) Hertling defined the Picard-Fuchs singularity in [Hel] as follows:
Definition The Hertling invariant Her\(f) of a singularity / is the set
i.e. the 'orbit' of its Brieskorn lattice in .Ж wrt the group Fi.
E.2.6) Lemma The two invariants of a singularity /, the Picard-Fuchs
singularity PFS{f) and the Hertling invariant Her\ (/), coincide.
This is almost a tautology, since both these invariants are given by the
period matrix Q(t) to within the equivalence relation. If y(-^°\y))
€ Her\(f), then choosing a basis s\, ..., s^ in y(^i0\y)) and a basis
6], ...,6ц in the space H* of multivalued horizontal sections of the
5 The Picard-Fuchs singularity and Hertling's invariants 177
holomorphic fibration H^,, we obtain the matrix ((s,(J), <5/0))> which
determines an element of PFS{f).
E.2.7) Now if fv(x) is a /г-const family of isolated hypersurface singula-
singularities, parametrized by a manifold Y, we obtain the map
Ф,:Г-П/Г,,
and we can regard the Hertling invariant Her\ (fy) of the singularity fy(x)
as the point Ф\(у) е П/Г] in quotient space.
E.2.8) Denote by '—«' the equivalence relation on Y given by the in-
inequivalence relation of functions.
У\ ~уг ¦& the germs fyi (x) and fn{x) are /{-equivalent.
R
By E.2.3) we obtain the map
Ф,:Г/~-П/Г,
R
(we are not interested here in the structure of quotients as analytic spaces).
Now we can formulate the Varchenko's conjecture as follows.
Conjecture The map Ф) is (locally) injective.
5.3 The Hertling invariants Her2(f) and
E.3.1) As was shown by Hertling's calculations [Hel] for unimodular
singularities the local form of Varchenko's conjecture holds, but the global
form doesn't always.
Example Let us consider the /г-const deformation of the singularity E\i
from C.3.1). Let m0 = min(m\Tm = 1) = 42 be the order of the mono-
monodromy, and mR - mo/2 = 21. Let Ф: Y = С -> P1 = П, a i-» A: a), be
the monodromy map C.3.7). It is obvious that if (a\/a2)m« = 1, then fai
and fa2 are Л-equivalent. The converse is also true. Thus, /a, and /a, are
^-equivalent <?» (a\/a2)m* = 1, and at the same time, as Hertling showed,
PFS(fai) = PFS{fai) «¦ №-
=: 1.
Consequently, for the singularity E\2 the period map Ф1: Y/ ~ —> ГТ/Г1 is
two-sheeted, for every Picard-Fuchs singularity there are exactly two in-
inequivalence classes of singularities with given PFS.
E.3.2) Thus, the invariant PFS(f) = Her\{f) is sufficiently rough, in the

178
/// The period map of a fi-const deformation
sense that it does not separate the Л-equivalence classes. This led Herding
to consider finer invariants, which were, as before, invariants of R-
equivalence classes, but which separated the Л-equivalence classes. We
have to consider subgroups of G\ which respect more topological proper-
properties than G\ does. For g* € GL(Hn(Xx, Z)) we denote by
g* € GL(H"(XX, Z)) the automorphism of the dual Milnor lattice
H\XX, I).
Put
G2 = {g* € G\\g* preserves the intersection form on Н„(ХХ, Ж)},
Gi — {g* € G\g* preserves the bilinear Seifert form}.
Since (see [AGV]) the Seifert form determines the intersection form and
the monodromy, we have G\ D G2 D G3. Denote by
Г3 СГ2 С Г, с Aut.-Ж
the corresponding groups of automorphisms of the meromorphic connec-
connection .Ж.
E.3.3) Definition Hertling defined the invariants Нег,(/), i = 1, 2, 3, of
a singularity / as the sets
Her,(f) = {у(.Ж@))|у € Г,},
i.e. the 'orbit' of its Brieskorn lattice in. /6 wrt the group Г,-.
Thus, the invariants //er,(/), i= 1,2,3, are the invariants of R-
equivalence classes of increasing fineness,
jg(f) = .?(g) => Her3(f) = Her3(g) => Her2(J)
= Her2{g) => Heriff) = Hen{g).
From the point of view of the period map of a family of singularities
fy(x), we can consider the following commutative diagram of quotients
and maps
Y -~
П/Г,
П
П/Гз
We can regard the Hertling invariant Яег,(/У) of the singularity fy(x) as
the point Ф,0>) € П/Г,-, i = 1, 2, 3, in the quotient space of the period
space.
5 The Picard-Fuchs singularity and Hertling's invariants 179
5.4 Hertling's results
E.4.1) In [Hel] and [He2] Hertling considered the invariants Her\ and
Her2 for all uni- and bimodal singularities. He proved the following
theorem.
Theorem
(i) The invariant Her2 separates the Л-equivalence classes (i.e. Ф2 is
injective) for all uni- and bimodal singularities, possibly with the excep-
exceptions of the series .Zi.ub ?i,iob Sf ]Ok, к € M, in the classification of
Arnol'd (see [AGV]).
(ii) The invariant Her\ determines the Л-equivalence classes for the
simple-elliptic singularities E6, ?7, E%, for the hyperbolic singularities
Tp,q,r and for the exceptional unimodal singularities ?13, Z\\, Z\2, Z^,
Qu, W\-$, S\\,S\2.
(iii) For the remaining exceptional singularities, E\2, E^, Q\o, Q\2, W\2,
U\2, there are two Л-equivalence classes of singularities with equal
invariants Her\.
For the series Z\^k, S\^Ok, SflOk the methods used are not sufficient to
calculate Her2 explicitly. However, Hertling believes that Her2 also
determines the Л-equivalence classes for them.
Hertling's results are formulated in more detail for the unimodal
singularities as follows:
E.4.2) For the 14 exceptional unimodal singularities we can take the fi-
const deformation fa(x) in the normal form
fa = fo(x, y, z) + aM,
where /0 is a quasihomogeneous polynomial of weight 1, wt(M)>l,
aeC.
Let m0 be the order of monodromy T, and
{mo, if mo is odd,
m0 ..
—, if m0 is even.
Two singularities/Ol and/02 are Л-equivalent, i.e. -f?,(fa^) = ,rt(fai) if and
onlyif(ai/a2)m* = 1.
Theorem
(i) Her2(fai) - Her2(fa2) «• @1/02)"* = 1 & -Щ/.х) = Mf*d-
(ii) For eight of exceptional unimodal singularities ?13, Z\\, Z\2, Z13,

180 III The period map of a fi-const deformation
Qw, Wn,SiuS\z we have
/аЛ*
Hen(fai) = Hen(fa2) «¦ f-ij = 1.
(iii) For the singularities ?12, ?14, Q\o, Qn, W\2, and U\2 we have
Яег,(/-в1) = Herx(Ja2) «¦ №\ ""= 1.
E.4.3) For the hyperbolic singularities Tp,q,r consider the //-const defor-
deformation given in the normal form C.4.1).
Theorem
Herx<Jax) = Herx{fai)&[-±
\cm(p,q,r)
E.4.4) The simple-elliptic singularities E6, E7, E% can be given in, for
example, the Legendre normal form, see C.5.2). The Л-equivalence class
of such a singularity is determined by the ./-invariant of the corresponding
elliptic curve.
Theorem
Hen(fXl) = Herx(fh) «¦ /A,) = /(A2) «¦ ./%fly) = jg{fXl).
E.4.5) The results obtained by Hertling enabled him to modify Varchen-
ko's conjecture as follows:
Conjecture The invariant Her2 (or Her{) of an isolated hypersurface
singularity determines its Л-equivalence class.
The methods applied by Hertling, which use the classification of
singularities and explicit calculations (which even in the case of bimodal
singularities are very cumbersome and difficult), cannot be applied to the
proof of this conjecture. It is not yet clear how to prove this conjecture in a
general form.
References
[A] A'Campo: Invent. Maths, 1973, 20, 147-69.1
[Ar] Arnol'd, V I.: On some problems in singularity theory. Geom. and Anal.,
papers dedicated to the memory of V. K. Patody, 1981,1-10.
[AGV] Arnold, A. I., Gusein-Zade, S. M., Varchenko, A. N.: Singularities of ¦
Differentiable Maps, Vols. 1 and 2, Monographs in Mathematics 82 and
83, Birkhauser, Boston, 1985 and 1988.
[Br] Brieskorn, E.: Die Monodromie der isolierten Singularitaten von Hyper-
flachen, Manuscripta Math., 1970,2, 103-61.
[Br-k] Brieskorn, E., Rnorrer, H.: Ebene algebraische Kurven, Birkhauser,
1981.
[B-Z] Brylinski, J.-L., Zucker, S., An overview of recent advances in Hodge
theory, Encyclopaedia of Math. Sciences, VINITI, 1991, vol. 69,
pp 48-165.
[B-G] Buchweitz-Greuel, Invent. Maths, 1980, 58,241-81.
[C-L] Coddington and Levinson, Theory of Ordinary Differential Equations,
McGraw Hill, New York, 1955.
[Da] Dam'lov, V I.: Newton polyhedra and vanishing cohomology, Funct.
Anal. Appl, 1979,13, No. 2, 32-47 (in Russian)
[Dl] Deligne, P.: Equations Differentielles a Points Singular Regulier, Lect.
Notes in Math., Springer, Berlin, 1970, vol. 163.
[D2] Deligne, P.: Theorie de Hodge, II, Publ. Math. IHES, 1971,40, 5-58.
[D3] Deligne, P.: Theorie de Hodge, III, Publ. Math. IHES, 1974, 44, 5-77.
[D-Di] Deligne, P., Dimca, A.: Filtrations de Hodge et par l'ordre du pole pour
les hypersurfaces singularities, Ann. scient. Ec. Norm. Sup., 4 serie.
1990, 23, 645-56.
[Di 1 ] Dimca, A.: On the Milnor filtrations of weighted homogeneous poly-
polynomials, Compos. Math., 1990, 76, No 1-2,19-47.
[Di2] Dimca, A.: Differential Forms and Hypersurface Singularities, Lect.
Notes Math., Springer, Berlin, 1991, No 1462, pp. 122-53.
[Di3] Dimca, A.: Singularities and Topology of Hypersurfaces, Springer
Verlag, 1992,263pp.
[DS] Doom, van, M. G. M., Steenbrink, J. H. M.: A supplement to the
monodromy theorem, Abh. math. Semin. Hamburg, 1989, 59,225-33.
[E] Elkik, R.: Singularites rationnelles et deformations, Invent. Math., 1978,
47,139-48.
181

182 References
[G-M] Gelfand, S. I., Manin, Yu. I.: Homological Algebra, in series Itogi nauki
i techniki, seria Seuremennie problemy matematiki, VINITI, 1989, vol.
38, 238 pp.
[Go] Goryunov, V V Adjacencies of spectra of certain singularities, Vestnik
MGU, ser math., 1981, 4,19-22.
[GI] Greuel, G. -M.: Der Gauss-Manin-Zusammenhang isolierter Singular-
itaten von vollstandigen Durchschnitten, Math. Ann. 1975, 214, 235-
66.
[Gr] Griffiths, P.: On the periods of certain rational integrals I, II, Ann. Math.,
1969,90,460-541.
[H] Hamm, H.: Zur analytischen und algebraischen Beschreibungder
Picard-Lefschetz-Monodromie, Habilitationsschrift, Gottingen, 1974,
I06pp.
[Hel] Hertling, C: Analytische invarianten bei den unimodularen und bimodu-
laren Hyperflachensingularitaten. Dissertation Bonner Mach. Schriften
250, Bonn 1992.
[He2] Hertling, C: Ein Torelli-Satz fur die unimodalen und bimodalen Hyper-
Hyperflachensingularitaten, Math. Ann., 1995, 302, 359-94.
[Kal] Karpishpan, Ya.: Pole order filtration on the cohomology of algebraic
links, Compos. Math. 1999, 78, No 2, 213-26.
[Ka2] Karpishpan, Ya.: Torelli theorems for singularities, Invent, math., 1990,
100,97-141.
[K-Ku] Kulikov, Vic. S., Kurchanov, P. F.: Complex algebraic varieties: periods
of integrals, Hodge structures. Encyclopaedia of Math. Sciences, 1989,
36,5-231.
[Ко] Kouchnirenko, A; Invent. Math., 1976, 32, 1-31.
[L] Le Dung Trang: Sur les noeuds algebriques, Compos. Math., 1972, 25,
281-321.
[L-R] Le Dung Trang, Ramanujam, С R: The invariance of Milnor's number
implies the invariance of topological type, Amer. J. Math., 1976, 98,
67-78.
[M1 ] Malgrange, В.: Integrates asymptotiques et monodromie, Ann. Sci Ecole
Norm. Super, 1974, 7, 405-430.
[M2] Malgrange, В.: Letter to the editors, Invent. Math, 1973, 20, 171-2.
[Mi] Milnor, J.: Singular Points of Complex Hypersurfaces, Annals of Math.
Studies, Princeton Univ. Press, Princeton, 1968, vol 61.
[N] Navarro, A.: Sur la theorie de Hodge-Deligne, Invent. Math., 1987, 90,
11-76.
[Phi] Pham, F.: Singularites des Systemes Dijferentiels de Gauss-Manin,
Progress in Math., Birkhauser, Boston, 1979, vol. 2.
[Ph2] Pham, F.: Structures de Hodge mixtes assosiees a un germe de fonction a
point critique isole, Asterisque, 1983, No 101-102, 268-285.
[SK] Saito, K.: Period mapping associated to a primitive form, Publ. Res. Inst.
Math. Sci., 1983,19, No 3, 1231-64.
[Sal] Saito, M.: Gauss-Manin system and mixed Hodge structure, Proc.
Japan Acad., 1982, 58, Ser. A, 29-32.
[Sa2] Saito, M.: Supplement to 'Gauss-Manin system and mixed Hodge
structure', Asterisque, 1983, No 101-102, 320-31.
[Sa3] Saito, M.: On the exponents and the geometric genus of an isolated
hypersurface singularity, Proc. Symp. Pure Math. AMS, 1983, 40, Pt. 2,
465-72.
References
183
[Sa4] Saito, M.: Hodge filtrations on Gauss-Manin systems. I, J. Fac. Sci
Univ. Tokyo, 1984, Sec. IA, 30, No 3,489-98.
[Sa5] Saito, M.: Hodge structure via filtered D-modules, Asterisque, 1985 No
130,342-51.
[Sa6] Saito, M.: Exponents and Newton polyhedra of isolated hypersurface
singularities, Math. Ann., 1988, 281, 411-17.
[Sa7] Saito, M.: On the structure of Brieskorn lattice, Ann. Inst. Fourier, 1989,
39, No 1, 27-72.
[Sa8] Saito, M.: Period mapping via Brieskorn modules, Bull. Soc. Mat. Fr,
1991,119, No 2, 141-71.
[Sa9] Saito, M.: On Steenbrink's conjecture, Math. Ann., 1991, 289, No 4,
703-16.
[SalO] Saito, M.: Mixed Hodge modules and applications, Proc. Int. Congr.
Math., Kyoto, Japan, 1990. Math. Soc. Japan, 1991, 725-34.
[Sci] Scherk, J.: On the Gauss-Manin connection of an isolated hypersurface
singularity, Math. Ann., 1978, 238, 23-32.
[Sc2] Scherk, I: On the monodromy theorem for isolated hypersurface singu-
singularities, Invent. Math., 1980,58, 289-301.
[Sc3] Scherk, J.: A note on two local Hodge filtrations, Proc. Symp. Pure Math.
A.M.S., 10, 1983, Part 2, 473-7.
[Sc4] Scherk, I: Isolated singular points and the Gauss-Manin connection,
PhD thesis, University of Oxford, 1977.
[Sc-S] Scherk, J. and Steenbrink, J. H. M.: On the mixed Hodge structure on the
cohomology of the Milnor fibre, Math. Ann., 1985., 271, No 4, 641-65.
[Sm] Schmid, W.: Variation of Hodge structure: The singularities of the period
mapping, Invent. Math., 1973, 22, 211-319.
[Se] Sebastiani, M.: Preuve d'une conjecture de Brieskorn, Manuscr. Math.,
1970, 2, No 2, 301-6.
[Se-T] Sebastiani, M. Thom, R.: Un resultat sur le monodromie. Invent. Math.,
1971,13, No 1-2, 90-6.
[Si] Siersma, D: The monodromy of a series of hypersurface singularities,
Comment. Math. Helv., 1990, 65, No 2, 181-97.
[SI] Steenbrink, J. H. M.: Limits of Hodge structures, Invent. Math., 1975,
31, No 3,229-57.
[S2] Steenbrink, J. H. M.: Intersection form for quasihomogeneous singula-
singularities, Compos. Math., 19П, 34, No 2, 211-23.
[S3] Steenbrink, J. H. M.: Mixed Hodge structure on the vanishing cohomol-
cohomology, in Real and Complex Singularities, Oslo 1976, Alphen aan den Rijn,
Oslo 1977, pp. 525-63.
[S4] Steenbrink, J. H. M.: Mixed Hodge structures associated with isolated
singularities, Proc. Symp. Pure Math. AMS, 1983, 40, Part 2, 513-36.
[S5] Steenbrink, J. H. M.: Semicontinuity of the singularity spectrum, Invent.
Math., 1985, 79, No 3, 557-65.
[S6] Steenbrink, J. H. M.: The spectrum of hypersurface singularities, Aster-
Asterisque, 1989, No 179-80, 163-84.
[S7] Steenbrink, J. H. M.: Applications of Hodge Theory to Singularities,
Proc. Int. Congress of Math., Kyoto, Japan, 1990. Math. Soc. Japan,
1991, pp. 569-76.
[Sr] Straten van D.: On the Betti numbers of the Milnor fibre of a certain
class of hypersurface singularities. Lect. Notes Math., Springer, Berlin,
1987, No 1273, pp. 203-20.

184 References
[V1 ] Varchenko, A. N.: Gauss-Manin connection of isolated singular point
and Bernstein polynomial, Bull. Sci. math., 2nd series, 1980,104,
205-23.
[V2] Varchenko, A. N.: Asymptotics of holomorphic forms define mixed
Hodge structure, Dokl. Acad. NaukSSSR, 1980, 225, No 5, 1035-38
(in Russian) [English transl.: Sov. Math. Dokl, 1980, 22, 772-5].
[V3] Varchenko, A. N.: Asymptotic Hodge structure in vanishing cohomo-
logy, Izv. Akad. NaukSSSR, Ser. Mat., 1981, 45, 540-91 (in Russian)
[English transl.: Math. USSR Izv., 1982, 18,469-512].
[V4] Varchenko, A. N.: On the monodromy operator in vanishing cohomology
and the operator of multiplication by /in the local ring, Dokl. Acad.
Nauk SSSR, 1981, 260, No 2, 272-6 (in Russian) [English transl.: Sov.
Math. Dokl, 1981, 24, 248-52].
[V5] Varchenko, A. N: The complex exponent of a singularity does not
change along strata /x =const, Fund. Anal. Appl., 1982,16, No 1,1-12.
[V6] Varchenko, A. N: A lower bound for the codimension of strata ц =const
in terms of the mixed Hodge structure, Vestnik MGU, Mat. Mech., 1982,
No 6, 28-31 (in Russian) [English transl.: Moscow Univ. Math. Bull,
1982,37,30-3].
[V7] Varchenko, A. N.: The spectrum and falling to pieces of a critical point
of a function, Dokl. Acad. NaukSSSR, 1983, 270, No 2, 267-70
(in Russian).
[V8] Varchenko, A. N.: On semicontinuity of the spectrum and an upper
bound for the number of singular points of projective hypersurfaces,
Dokl. Acad. NaukSSSR, 1983, 270, No 6, 1294-7 (in Russian).
[V9] Varchenko, A. N.: Asymptotics of integrals and Hodge structures, in
Modern Problems of Mathematics, VINITI, 1983, vol. 22, 130-66
(in Russian).
[V10] Varchenko, A. N.: On the local residue and the intersection form on the
vanishing cohomology, Izy. Akad. NaukSSSR, Ser. Mat., 1985, 49, No 1,
32-54 (in Russian) [English transl.: Math. USSR Izv., 1986, 26, 31-52].
[V-Ch] Varchenko, A. N, Chmutov, S. V: Tangent cone to the stratum /г =const,
Vestnik MGU, Mat. Mech., 1985, No 1, 6-9 (in Russian) [English transl.:
Moscow Univ. Math. Bull, 1985, 40, 7-12].
[V-Kh] Varchenko, A. N., Khovanskii, A. G.: Asymptotics of integrals over
vanishing cycles and the Newton polyhedron, Dokl. Acad. Nauk SSSR,
1985, 283, No 3, 521-5 (in Russian) [English transl.: Sov. Math. Dokl,
1985,32,122-7].
[Y] Yomdin, I. N.: Complex varieties with singularities of dimension one,
Siberia Math. 1,1974,15, 1061-82.
Index
clj = -(\/ln\)\nXj A.9.4.1),
A1.6.1.6L9,96
Bmx (U.4.3.3) 85
cohomology
DeRham A.3.2.3) 11
Koszul A.4.3.2) 18
commutator rule A.7.7.5) 35
connection
canonical A.2.5.3) 7
coefficients A.2.3.5) 6
dual A.2.6.1) 8
Gauss-Manin A.1.3.1J
in the pair q.4.4.3) 20
meromorphic A.4.5.1) 20
on a sheaf A.2.1.1K
regular singular A.7.7.1) 39
covariant derivative A.2.2.1) 4
covariant differentiation A.2.1.1) 4
De Rham cohomology A.3.2.3) 11
De Rham complex A.2.4.4) 7
decomposition
block A1.6.1.1)95
of a meromorphic connection A1.6.2.5) 97
root A1.6.2.4) 97
degree of spectrum A1.8.9.3) 130
elemenrary section s[A, a] A1.6.4.5) 100
extension
canonical A1.2.6.4), A1.6.3.7) 69, 98
fibre
canonical A1.2.4.6), A1-2.4.8) 65, 66
Milnor A.1.1.1I
filtration
Hodge A1.1.1.2), A1.1.3.1N0,61
limit Hodge according to '
Scherk-Steenbrink A1.7.2.1),
A1.7.3.2) 106, 108
Schmid A1.2.5.8) 68
Steenbrink A1.3.2.4), A1.3.4.4) 76, 79
Varchenko A1.7.5.1) 111
Newton A1.8.5.3) 122
of a nilpotent operator A1.2.7.1) 71
on a quotient A1.8.4.5) 120
pole order A1.4.2.5), A1.4.5.1) 84, 87
V*- A1.6.3.3) 98
weight A1.1.3.1) 61
framed system of linear differential
equations A11.5.1.3) 173
Gauss-Manin connection A.1.3.1) 2
Gauss-Manin differential system
A1.5.2.3) 90
Gysin exact sequence (II.4.1.1) 82
//A.1.2.1J
Hx A.1.2.1J
.Ж A.1.3.1J
Жх A1.5.2.3) 90
.JT<-2» = .XK(X/S) A.3.3.4),
A.5.3.2I2,26
(-i) = '._#¦ A.4.4.1), A.5.3.2) 19, 26
= «ж A.5Д.З), A.5.3.2) 24,26
Hertling's invariants A11.5.2.5), A11.5.3.3)
176,178
Hodge
filtration A1.1.1.2), A1.1.3.1) 60, 61
numbers A1.1.3.2) 61
structure A1.1.1.1) 60
polarized A1.1.2.1) 61
Jacobi algebra = Milnor algebra A.4.3.5) 18
join of singularities A1.8.7.1) 127
185

186
Index
key-lemma A.7.6.2) 38
Koszul cohomology A.4.3.2),
A.10.7.1) 18,57
Koszul complex A.4.3.1) 17
lattice
Brieskorn A.5.1.3) 24
canonical A1.6.3.6) 98
in a vector space A.4.5.5) 20
saturated A.7.7.2) 39
leading part of cu € . Ji A1.6.3.2) 98
Leibniz identity A.2.1.1K
Leray operator A.4.1.5) 15
Leray residue theorem A.4.1.6) 16
limit
MHS A1.2.8.1), A1.3.2.7) 73, 76
monodromy A.7.8.1) 41
.Л A1.6.1.1)95
MHS
morphism A1.1.3.3) 62
-period map A11.3.6.5) 164
Milnor algebra A.4.3.5) 18
Milnor cohomological fibration A.1.2.1),
A.2.6.1J,8
Milnor fibration A.1.1.1) 1
Milnor fibre A.1.1.1) 1
Milnor homological fibration A.1.2.1),
A.2.6.1J,8
Milnor number A.1.4.5), A.10.1.1) 3, 51
monodromy
horizontal (II.8.11.9), (Ш.5.2.1) 137, 175
logarithm of the unipotent
part A1.2.5.6) 68
theorem A.9.1.1), A1.3.5.9) 47, 81
supplement A1.3.5.10) 81
transformation A.1.4.4) 3
vertical (II.8.11.9) 137
N = (l/2m)log Tu (II.2.5.6) 68
order
fimction A1.8.4.5) 120
function a A1.6.3.1) 98
Newton A1.8.4.1), A1.8.5.2) 119, 122
of a geometric section A11.2.4.2) 150
of an elements 6 .Ж(И.6.3.1) 98
period
map A1.2.3.4) 64
defined by embedding of Brieskorn
lattices A11.3.2.5) 153
defined by MHS in vanishing
cohomology A11.3.6.5) 164
MHS A11.3.6.5) 164
matrix A.8.1.6) 43
of a differential form A.8.1.1) 42
space A1.2.3.3) 64
space for Hodge filtrations П
A11.3.6.4) 164
space П A11.3.2.8) 154
space П' A11.3.2.5) 153
Picard-Fuchs equation A.8.1.5), A11.5.1.1)
43, 172
Picard-Fuchs singularity A11.5.1.5) 174
regularity theorem A.8.2.1) 44
residue
Leray theorem A.4.1.6) 16
of a meromorphic connection A.7.7.4) 39
of a system A.7.5.1) 37
root
decomposition of. Ж A1.6.2.5) 97
component of order a A1.6.2.6) 97
section
elementary of order a A1.6.4.5) 100
geometric A.5.1.6), A11. 2.1.2) 25, 145
horizontal A.2.3.7) 6
semicomtinuity domain A1.8.9.7) 131
sheaf of
nearby cycles A1.8.11.8) 136
vanishing cycles A1.8.11.8) 136
singular point of a differential equation
regular A.7.4.1) 36
simple A.7.5.1) 36
spectral multiplicities (II.8.1.4) 116
spectral numbers A1.8.1.4) 116
spectrum
of a singularity A1.8.1.4),
A1.8.10.4) 116, 133
pairs A1.8.2.1) 117
properties A1.8.3.2), A1.8.3.3) 118
Torelli theorem infinitesimal A11.4.3.1) 169
variation of Hodge structure A1.2.2.1) 63
Yomdin series A1.8.11.1) 134