Page i
m
P IXK If.
Complex Algebraic Geometry
An Introduction to Curves and Surfaces
Kichoon Yang
Arkansas State University
State University, Arkansas
MAitCEL Dekker, Inc. New York ' Basel - Hong Kong
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page ii
Library of Congress Cataloging-in-Publication
Yang, Kichoon.
Complex algebraic geometry: an introduction to curves and
surfaces / Kichoon Yang.
p. cm. — (Monographs and textbooks in pure and applied
mathematics)
Includes bibliographical references and index.
ISBN 0-8247-8591-6 (acid-free paper)
1. Geometry, Algebraic. 2. Functions of complex variables.
I. Title. II. Series.
QA564.Y36 1991
516.3'5-dc20 91-22527
CIP
This book is printed on acid-free paper.
Copyright © 1991 by MARCEL DEKKER, INC. All Rights Reserved
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or
mechanical, including photocopying, microfilming, and recording, or by any information storage and
retrieval system, without permission in writing from the publisher.
MARCEL DEKKER, INC.
270 Madison Avenue, New York, New York 10016
Current printing (last digit):
10 98765432
PRINTED IN THE UNITED STATES OF AMERICA
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page iii
To the memory of my grandmother
Okji Yang
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page V
Preface
The common zero locus of a system of polynomial equations is called an algebraic variety; algebraic
geometry, roughly speaking, is the study of algebraic varieties. Although algebraic geometry is amongst the
oldest branches of mathematics it has remained a relatively inaccessible discipline. The reasons for this are
two-fold: algebraic geometry is intimately connected with many other areas of mathematics such as several
complex variables, commutative algebra, and topology; algebraic geometry has undergone a sort of
metamorphosis in recent years to attain the Hilbertian rigor.
The present book represents a modest attempt to introduce various aspects of algebaric geometry,
emphasizing the transcendental aspect. The exposition is at the second year graduate level: it assumes some
familiarity with algebraic topology, function theory, and elementary differential geometry. Properly
speaking, the book consists of a collection of seminar notes, and is designed as a guide to complex algebraic
geometry for the nonexpert.
Chapter I contains preliminary materials from commutative algebra and algebraic topology. Chapter II deals
with various complex-analytic techniques, and our exposition was strongly influenced by P. Griffiths and J.
Harris, Principles of Algebraic Geometry (Wiley, 1978). Chapter III contains an exposition of algebraic
curves and compact Riemann surfaces. The Enriques-Kodaira classification of algebaric surfaces is given in
Chapter IV. The last chapter deals with Hermitian differential geometry and its possible ramifications in
complex algebraic geometry: a differential geometer may find this chapter interesting. Each chapter comes
with its own introduction explaining the organization and citing further references, which can be found in the
Bibliography.
Finally I would like to thank Bob Fisher, Jim Glazebrook, and Gary Jensen: they read various parts of the
manuscript and made helpful suggestions and comments. In particular, my thanks go to Bob Fisher who has
kindly contributed the appendix, "Some Background on the Linear Algebra of Complex Forms." I also want
to thank Andy Talmadge for preparing the index. This writing project was supported in part by a grant from
Arkansas State University.
KICHOON YANG
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page vii
Contents
Preface v
Chapter I. Algebraic and Topological Preliminaries j_
§ 1. Affine and Projective Varieties j_
§ 2. Ideals and Varieties 4
§ 3. Irreducible Ideals and the Coordinate Ring 9
§ 4. Analytic Varieties j^
§ 5. Dimension Theory j^
§ 6. The Degree 23
§ 7. Simplicial Homology and the Intersection Number 26
§ 8. De Rham Cohomology and Poincare Duality 3]_
§ 9. The Hodge Theorem 35
Chapter II. Transcendental Methods 44
§ 1. Sheaves and Cohomology 44
§ 2. Complex Vector Bundles and the Chern Class 5]_
§ 3. Line Bundles, Divisors, and Linear Systems 63
§ 4. Hodge Theory for Kahler Manifolds 74
§ 5. Bundles over Complex Projective Space 84

Chapter III. Curves and Compact Riemann Surfaces 92
§ 1. Plane Curves 93
§ 2. Meromorphic Functions and Meromorphic Forms 108
§ 3. Linear Systems of Divisors 112
§ 4. The Jacobian Variety and Abel's Theorem 122
§ 5. Weierstrass Points and Hyperelliptic Riemann Surfaces 134
§ 6. Projective Embeddings 144
Chapter IV. Algebraic Surfaces and the Enriques Classification 151
§ 1. The Intersection Pairing 152
§ 2. Rational Maps and the Blow-up 163
§ 3. The Kodaira Dimension 171
§ 4. Ruled Surfaces 176
§ 5. Rational Surfaces 183
§ 6. The Enriques Classification 190
§ 7. K-3 Surfaces 199
§ 8. General Type Surfaces 210
§ 9. Complex Spaces and Singular Surfaces 214
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page viii
Chapter V. Hermitian Differential Geometry 220
§ 1. Grassmannians 221
§ 2. Space Curves: The Plucker Formulae 232
§ 3. Complex Submanifolds: Weyl's Formulae 239
§ 4. Projective Hypersurfaces and Their Chern Numbers 248
§5. Surfaces in f« 259
Appendix I. Some Background on the Linear Algebra of Complex 267
Forms
Robert Fisher
§ 1. Complexification 267
§ 2. Complex Forms 271
§ 3. Complex (l,l)-forms 277
Appendix II. Elliptic Functions 280
Bibliography 289
Index 297
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Chapter I. Algebraic and Topological Preliminaries
In this chapter we give an introduction to varieties; we review some
materials from Algebraic Topology.
Although our approach in this book is primarily complex-analytic in flavor
it is essential to have a good grasp of basic algebraic notions: prime ideals and
irreducible varieties, the coordinate ring and function field of an irreducible
variety, transcendental field extensions, and so on. Proofs of foundational results
such as the Hilbert Basis Theorem and Hilbert Nullstellensatz are also given in
this chapter. A good algebraic introduction to varieties can be found in [K] or
[F]. A comprehensive treatment is found in [H].
Assuming some familiarity with Several Complex Variables we give a quick
introduction to analytic varieties in §4. The reader may consult [GR] or [GH]
for futher materials.
§§7-8 deal with the smooth simplicial homology and de Rham cohomology
groups of a compact oriented differentiable manifold. Poincare duality and an
important concept, namely the Poincar^ dual of a closed submanifold, are
discussed. An elementary account of the materials covered in these two sections
can be found in [ST]. Finally in §9 a proof of the Hodge theorem for a compact
oriented Riemannain manifold is given, setting the stage for Hodge theory for
Kahler manifolds to be given in the next chapter.
§1. Affine and Projective Varieties
An affine algebraic (or simply afflne) variety is a subset of C° which can be
realized as the common zero locus of a collection of polynomials in C[x ,'",x°],
the polynomial ring over C with n variables.
P", the complex projective space of dimension n, is the space of complex

lines in C"''"^ Let
and define an equivalence relation by decreeing that
X ~ X if X = Ax for some A 6 C* = C\{0}.
Then IP° is identified with the quotient space C°''"^\{0}/C*. We use the notation
[x], or sometimes / , to denote the point in IP° determined by x. (x*), 0 < i < n,
are called the homogeneous coordinates of [x]. Fix j and put
C- = "tW 6 P": xJ = 0} ^ IP°-^
We then have
pi
\iP:;5 = {[y°.---.i.---.y"]:/ = x'M^c".
IP°\IP°i is called the affine part of IP° relative to the hyperplane at infinity iP°i.
(y ) • • • )y°) are called the inhomogeneous coordinates. Note that
,ni inn-l\
IP° = u (IP°\IP°'!).
A homogeneous subset S c C""*" is a subset with the property that
whenever x 6 S, Ax 6 S for every A € C.
An affine variety in C is called a homogeneous variety if it is also a
homogeneous set. It is easy to see that an affine variety is homogeneous if and
only if it can be realized as the zero locus of a collection of homogeneous
polynomials.
A homogenous set in i defines a set in IP°, and vice versa.
Definition. A projective algebraic (or simply projective) variety is a subset of IP°
given by a homogenous variety in C""*" .
Choose a IP°~ . This gives rise to an inclusion
i: C -» IP°.
Given an affine variety V c C° we define the projective completion of V, denoted
by V, to be the smallest projective variety in IP° containing i(V).

Example. Consider the irreducible polynomial in C[x,y] given by
f(x,y) = y^ - X n (x^ - k2).
k=i
We leave it as an interesting excercise to the reader to verify that the projective
completion in IP of the affine variety V(f) is topologically a torus with g handles.
Let f 6 C[x°,... ,x^\xJ+\.. • ,x°]. Write
where d is the degree of f and each nonzero f, is a homogeneous polynomial of
degree k. The homogenous polynomial of degree d given by
h(f) = (xVfo + {^f-\ + ••• + fa e c[xO,...,xV.-.x"l
is called the homogenization of f at x^. The following proposition is
straightforward.
Proposition. Let V be an affine variety in C° defined by polynomials
{f, g, ... }cC[xV..,xi-\x^+\...,x"].
We then have
a) V c IP° is represented by the zero locus of {h(f),h(g),...} in C""*"^;
b) V is the topological closure of i(V) in IP°, where i is the inclusion of C"
in IP° given relative to IP°"^ = IP°i.
° CD CD,J
Let V c IP° be a projective variety. We fix a IP°~^ = IP°i. The affine part
of V is simply V\IP°~ . A homogeneous polynomial in C[x,...,x°] can be
dehomogenized by setting x^ = 1. Suppose V C IP° is given by some collection of
homogeneous polynomials in C[x .•••,x°]. Then the affine part of V, V\IP°~\
coincides with the affine variety defined by the dehomogenization. The projective
completion of the affine part of V, however, is not necessarily equal to V, albeit
always contained in V. (One may lose a part of V lying in P" .) It is true
that given an affine variety the affine part of its projective completion is always
equal to itself.

Any intersection of affine varieties in C° is again an affine variety in C°. A
finite union of affine varieties is also an affine variety. The Zariski topology on
C" has as its closed sets affine varieties. The Zariski topology is coarser than the
usual topology: for example, the affine varieties in C are just C, 0, and finite sets.
The usual topology on C° can be recovered by letting the zero loci of smooth
n 2n
functions on C ^ IR generate the closed sets.
§2. Ideals and Varieties
As the set of all varieties in C° or IP° naturally forms a lattice we begin
with a brief review of lattice theory.
Definition. A partially ordered set (p.o. set, to be short) is a set S with a binary
relation <, called a partial order, such that for every x,y,z 6 S:
X < x;
if X < y and y < x, then x = y;
if X < y and y < z, then x < z.
A map
<p: (S, <)-(S',<')
is called a p.o. set map if
whenever x < y, (fi{x) <' (fi{y).
A p.o. set map ^: S -» S is called a closure map if
X < ^x) and (potp{x) = ^x), x 6 S.
A partially ordered set (S, <) is said to satisfy the ascending chain
condition if there does not exist an infinite chain of the form
Xj <X2< ••• <x^< ...,
where x < y means x < y and x ^ y. The set (S, <) is said to satisfy the
descending chain condition if there does not exist an infinite chain of the form

X, > X- > • • • > X > • • •.
12 n
A partially ordered set satisfies the ascending chain condition if and only if every
nonempty subset contains a maximal element. (An element x e U c S is called a
maximal element if for every y € U, x < y implies x = y. Similarly one defines
the notion of a minimal element of a subset.) A partially ordered set satisfies
the descending chain condition if and only if every nonempty subset contains a
minimal element.
The least upper bound of x,y 6 S is often written x V y. The greatest
lower bound of x,y 6 S is written x A y. Given a partially ordered set the least
upper bound and/or greatest lower bound of an arbitrary pair of elements may
not exist. However, if they do, then they are unique.
Definition. A lattice is a partially ordered set in which any two elements possess
a least upper bound and greatest lower bound. A lattice map is a p.o. set map
that preserves V and A. A lattice reversing map
V'rCS, <, V, A)-(S',<', V, A')
reverses the partial order and takes V to A and A to V, that is to say,
X < y implies ^y) <' ^x),
V<x V y) = v<x) A' ip{y),
¥<x A y) = ip{x) V ifiy).
A lattice reversing bijection will be called an anti-isomorphism.
Let (S, <, V, A) be a lattice and suppose we are given a closure map
<p: S -» S.
Put
S' = {x 6 S: X = ^x)}.
Define
X V y = V'(x V y), x A' y = x A y.
This makes (S', <, V, A') into a lattice, called the closed lattice associated with

(s, ^).
Definition. Let S be a lattice.
a) S is said to be distributive if for every x,y,z 6 S we have
X A (yAz) = (xAy) V (xAz), x V (yAz) = (xVy) A (xVz)
(the two conditions are in fact equivalent);
b) X 6 S is said to be V-irreducible if whenever x = yVz, x = y or x = z;
c) X € S is said to be A-irreducible if whenever x = yAz, x = y or x = z;
d) a representation
X = y^ V • • • V y (respectively, x = y A • • • A y )
is said to be irredundant if there are no proper subsets of {y,,*",y } having the
least upper bound (respectively, the greatest lower bound) equal to x.
We now give the fundamental theorem of lattice theory.
Theorem. Let S be a lattice. We then have the following:
a) if S satisfies the ascending chain condition, then any x € S has an
irredundant representation
X = y, A • • • A y
•'1 •'n
with the y.'s all A-irreducible;
b) if S satisfies the descending chain condition then any x € S has an
irredundant representation
X = y V • • • V y
with the y.'s all V-irreducible;
c) if, moreover, S is distributive, then the above representations are unique.
Proof. Assume that S satisfies the ascending chain condition. If x € S is
A-irreducible, then there is nothing to do. So we may assume that x is not
A-irreducible and write x = y,Ay-, where x < y^y,- We may similarly break up
the y.'s into y.iAy... This process must terminate after finitely many steps for
otherwise S would not satisfy the ascending chain condition. One makes the

resulting representation consisting of A-irreducible elements irredundant simply by
deleting as many of the irreducible elements as possible. This proves a). The
proof of b) is similar. To prove c) suppose
x = y, A'"Ay =z, A'"Az
•'1 •'n 1 m
are two irredundant representations into A-irreducibles. Now
y. > z, A • • • A z , 1 < i < n,
•'i - 1 m' - -
So
y. = y. V (z,A' • -Az ).
Using distributivity,
y. = (y.VzJ A ••• A (y.Vz ).
Since y. is A-irreducible this means that
y. = y. V z. for some j.
So y. > z,. By symmetry z. > y for some k. Thus y, < y.. As the
representation
X = yj A ... A y^
is assumed to be irredundant we must have y. = y.. This implies that y. = z,.
The rest is left to the reader, d
Let R = C[x ,• • -.x"], and also let 1 denote the set of all ideals in R. 1 is
made into a lattice as follows: for I,J € 1 define I V J to be
I + J = {x+y: X € I, y € J}
(this is the smallest ideal containing both I and J), and define
I A J = I n J.
Also use the set theoretic C (or write c) as < on 2.
Let V denote the set of all affine varieties in C°. Then V becomes a lattice
by setting
"c" = "<" "u" = "v" "n" = "a"
In fact, V is a distributive lattice.

Consider the map
$: V -» 2,
V H the largest defining ideal of V in R.
$(V) is the largest ideal in C[x .•••,x°] vrith the property that V is the common
zero locus of all polynomials in $(V). Ideals in the image Im($) c J are called
closed ideals. Given I 6 1 define the closure of I, denoted by I, to be the largest
defining ideal of the variety V(I), the zero locus of I. Then the assignment
2^2, IH r
is a closure map on 2, and T (the associated closed lattice) is precisely Im($).
We will see later that an ideal is closed if and only if it is its own radical.
To give a simple example, the principal ideal <x > c C[x] is not closed
since
V(<x>) = V(<x^>), <x^> C <x>, and <x^> f <x>.
The ideal <x> is closed since it is a maximal ideal.
Theorem A, The lattice V is anti-isomorphic to T.
Proof. The anti-isomorphism is given by
V H $(V).
It is easy to see that
V c W iff $(W) c *(V),
$(V U W) = $(V) n *(W),
$(v n w) = $(v) + $(w). D
An ideal in R = C[x , • • • ,x°] is called a homogenous ideal if it can be
generated by a set of homogenous polynomials. The following is the projective
version of the above correspondence.
Theorem. The lattice of projective varieties in IP° is naturally anti-isomorphic to
the lattice of closed homogeneous ideals in C[x ,• • -.x"].

§3. Irreducible Ideals and the Coordinate Ring
By a ring, unless otherwise stipulated, we will mean a commutative ring
with a multiplicative identity ^ 0.
A ring R is called a Noetherian ring if the lattice of ideals in R, denoted
by 1, satisfies the ascending chain condition. A basis for an ideal I is a
collection {a. € 1} such that
I = {E r.a.: r. € R, finite sums).
R is a Noetherian ring if and only if every ideal has a finite basis. There is the
famous
Hilbert Basis Theorem. If R is Noetherian, then so is R[x].
Proof. Let A be an ideal of R[x]. Put
I„ = {0} U {r € R\{0}: 3 f € A with f(x) = rx° + lower terms}.
If a € I \{0}, then there is a polynomial f(x) 6 A whose leading term is ax°. So
x-f(x) 6 A, and a 6 I j.
This gives rise to a chain
L C I, C • • • C I C
0 1 n
• •
Since R is Noetherian there exists an integer k such that I, = I, = •••. For
each i, 0 < i < k, pick a finite basis of I., say {a..,'",a. }. For each pair (i,j)
let f. .(x) be a polynomial of degree i whose leading coefficient a... We claim that
the finite set
{tj(x): 0 < i < k, 1 < j < n.}
is a basis for the ideal A c R[x]. To establish the claim let g € A with degree t,
say
g(x) = ax + lower degree terms.
We do an induction argument on t. If t = 0, then g = o £ I . Hence o is a
linear combination of o„,,-",o- . Now by definition f.. = o... We now
01* * Ono "^ Oj Oj
assume that t > 0 and t < k. Now a € I. So for some r.'s in R
t 1

o = r,o,, + ••• + r^ o. .
1 tl nt tnt
Put
h = g-(^ftl+-"+^ntW-
Since the leading term of the expression inside the parentheses is ox the degree
of h is less than or equal to t-1. So by induction h is a linear combination of
the f..'s. Therefore, so is g. We now consider the case t > k. So a € I = I,.
ij ' •* ~ t k
We can write
for some s.'s in R. Consider
1
Then as before deg(h) < t-1, and the induction does the rest, d
For example, consider
A = <n+x°: n 6 Tl'^> C C[x].
The ideal A is, in fact, a principal ideal: it can be generated by 1+x.
Definition. An ideal I in R is said to be irreducible if
whenever I = I fl Ig (I. ideals), I = I^ or I = I .
The fundamental theorem of lattice theory together with the Hilbert basis
theorem give
Proposition. Let I be an ideal of R = C[x ,• • ^x"]. Then I can be written as an
intersection of irreducible ideals (not uniquely, however).
Definition. An (affine or projective) algebraic variety V is said to be irreducible
if the following holds:
whenever V = V. U V (all varieties), V = V. or V = V .
Theorem. Let I be a closed ideal of R = C[x ,-",x°]. Then I can be written
uniquely as an irredundant intersection of closed irreducible ideals.
Proof. V is a distributive lattice anti^somorphic io T. a
One has the projective version of the above theorem simply by considering
10

closed homogenous ideals in C[x .•••,x°].
The following important theorem generalizes the fundamental theorem of
algebra.
Theorem (Hilbert Nullstellensatz). Let R = C[xV'^x"]. Then every proper
(meaning, f R) ideal of R has a zero in C".
Since every proper ideal of R is contained in a maximal ideal the above
theorem is equivalent to the statement: every maximal ideal c R has a zero € C°.
Proof of the Nullstellensatz. We will use the following result from algebra: If an
extension k[a.,'",a ] is a field, then each a. is algebraic over k, where k is any
field. Put
k = C , a. = X* + m,
where m is the given maximal ideal of C[x , • • • ,x°]. Observe that
C[x\...,x°]/m-C[aj,...,aJ
is a field. So each a. is algebraic over C, hence a. € C. Consider the
isomorphism given by
C[x\- ",x^]/m -^ C, x'+m h a..
Under this map each fern gets mapped to 0 € C. So f(aj,' • •,a ) = 0. D
Corollary. The set of maximal ideals of C[x,«",x°] is in natural bijective
correspondence with the set of points of C°.
Proof. A point (a.,*",a ) € C° corresponds to the maximal ideal
m= <(x^-aj),..-,(x°-aj>
in C[x\...,x"]. D
The following theorem is another version of the Nullstellensatz.
Theorem. Let I be an ideal in C[x , • • • ,x°]. Then
n p = n m,
pDI mDI
where the p's are prime ideals, and the m's are maximal ideals.
11

Let I be an ideal of C[x .•••,x°]. Then the closure of I is the intersection
of all maximal ideals containing I. To see this let V denote the affine variety
defined by I. Now V = U p, and p is the point variety corresponding to a
p€V
maximal ideal, m . Applying the lattice anti-isomorphism V -* V we see that I
is the intersection of the m 's. On the other hand, since every maximal ideal in
C[x , • • • ,x°] corresponds to a point in C° any m containing I must equal m for
some p € V.
Exercise. Let R = C[x ,• • ^x"]. The radical of I is defined to be
Rad(I) = {r € R: r* € I for some i € J"*"}.
Prove the following statements:
a) Rad(I) = 1;
b) if I is homogeneous, then so is Rad(I).
The following theorem states that prime ideals in C[x ,■ ■ ■,x°] correspond to
irreducible varieties in C°.
Theorem. An ideal I in C[x , • ■ • ,x°] is a prime ideal if and only if it is closed
and irreducible.
Proof. Let I be a closed and irreducible ideal in R = C[x ,---,x°], and assume
that it is not a prime ideal. So there exist a,b € R\I with ab € L Now
I+<a> ^ I, I+<b> i I, (I+<a>)-(I+<b>) = L
Let V, V , V, denote the affine varieties defined by I, I+<a>, I+<b>,
respectively. Applying the lattice anti-isomorphism
V U V. = V, V ^ V, V^ ^ V.
a b ' a "^ ' b
showing that V is not irreducible, contradicting the assumption that I is
irreducible. Conversely, let I be a prime ideal. I is closed since I is an
intersection of prime ideals. To show that I is irreducible suppose that
I = ij n i^, I, M. \i I.
12

Let a € lAl , b € I \L. Then a,b € I. But a € Ij implies ab € Ij, and b € Ij
implies ab € L. So ab € I. But this means that I could not be a prime ideal.
D
Consider
I = <x°> c C[x].
Then Rad(I) = <x>. In general, if f(x) 6 C[x,-'-,x°] is an irreducible
polynomial, then Rad(<f>) = <f>.
Let V = V(p) denote the irreducible affine variety given by a prime ideal p
in R = C[x ,• • •,x°]. We have the following
Definition. The integral domain R/p, also denoted by R^, is called the
coordinate ring of V. If V c IP° is an irreducible projective variety given by a
homogeneous prime ideal p, then the domain
C[xV.-,x"]/^
is called the projective coordinate ring of V.
R/p is isomorphic to the integral domain
€[yi,---,yj, yi = x'+p.
Conversely, given any integral domain of the form C[y.,-'-,y ], y. € R, one can
find a prime ideal p such that R/p is isomorphic to C[y.,- • •,y ].
The lattice of closed ideals in R^ is naturally embedded in the lattice of
closed ideals in R: a closed ideal in R„ is simply a closed ideal in R containing
the prime ideal p.
Definition. Let V c C°, W c C"^ be irreducible varieties. A bijection
f: V -* W
is called a (regular) isomorphism if there exist polynomial maps (i.e., the
component functions are polynomials)
ip-. c -»c", t c" -» c
such that (p\y = f, ^1^ = F. For irreducible projective varieties V c iP° and
13

W c P'", V is said to be isomorphic to W if there are affine parts
Vj c V, Wj C W with Vj = V, Wj = W
such that V. is isomorphic to W. as an affine variety.
Theorem. Let V and W be irreducible affine varieties. Then they are isomorphic
to each other if and only if their coordinate rings are isomorphic to each other.
For an elementary proof of the above result we refer the reader to [K] pp.
140-141.
Let V = V(p) c C° be an irreducible variety. The quotient field, k„, of
the coordinate ring R^ is called the function field of V. For an irreducible
projective variety V c IP° the function field k„ is defined to be the quotient field
of the coordinate ring of any affine representative V.. (The reader should check
that this is well-defined.)
The field of rational functions on P" is given by
{0} U {^: f,g € C[x°,...,x°], g ^ 0, both homogeneous, deg(f) = deg(g)}.
This field is isomorphic to the quotient field, C(x ,'««,x°), of C[x ,---,x°]. The
function field k„, where V is an irreducible projective variety in IP°, is naturally
identified with the restriction of rational functions on IP° to V. It turns out that
a very useful notion of equivalence among algebraic varieties is that of birational
equivalence: two irreducible varieties are said to be birationally equivalent if their
function fields are isomorphic to each other. The birational classification of
varieties is far coarser than the classification up to regular isomorphism.
In the following we touch upon the concept of an abstract variety.
Let p be a prime ideal in R = C[x , • • • ,x°]. Also let V(p) denote the
irreducible variety consisting of the zeros of polynomials in p. Points of V(p)
correspond to maximal ideals of IL., or what is the same, maximal ideals of R
containing p. So
(Cj,- • ■,cj € V(p) if and only if p c <(x^-Cj),- • •,(x°-c )>.
14

Definition. Any integral domain D finitely generated over C is called an abstract
coordinate ring. The abstract variety V(D) is the set consisting of all maximal
ideals of D.
Let D be an abstract coordinate ring. By a representation of D we will
mean an isomorphism
D-C[yj,-..,yJ.
Fix a representation and consider the map
D = C[yj,..-,yJ-*C
induced by evaluating (yj,"-,y ) at (Cj,'",c ) € C°. The point c = (c.) € C° is
called an evaluation point of D if the above map is a ring homomorphism. The
set of all evaluation points of D in C° forms an irreducible variety V. So
V = V(p)
for some prime ideal p in C[x .•••,x°], and we have an isomorphism
C[x\...,x>-D.
The variety V c C° is called a model of the abstract variety V(D), and it is not
too difficult to show that all models of V(D) are isomorphic to each other.
§4. Analytic Varieties
Definition. A subset V of an open set U C C° is called an analytic variety in U
if for every x 6 V there is a neighborhood U c U of x such that the intersection
u n V
X
is the common zero locus of a finite set of holomorphic functions on U . A
subset V of a complex manifold M is called an analytic variety in M if it is
locally an analytic variety. (A coordinate neighborhood of M is identified with
an open set of C°.) An analytic variety V c M is called an analytic hypersurface
if it can be locally given as the zero locus of a single, not identically zero,
15

holomorphic function.
An analytic variety V in M, V ^ M, is closed, and M\V is dense and
connected.
A Weierstraaa polynomial in wis a. polynomial of the form
w + aj(zj,.--,z^_j)w "^ + ■■■ + aj(zj,---,z^_j), a.(0) = 0.
There is the famous
Wderstrass Preparation Theorem. If f is holomorphic about the origin in C° and
is not identically zero on the w-axis, then in some neighborhood of the origin f
can be written uniquely as the product f = g-h, where g is a Weierstrass
polynomial in w and h(0) ^ 0.
A geometric interpretation of the Weierstrass preparation theorem can be
given as follows: The zero locus of an analytic function f(z*,w), not vanishing
identically on the w-axis, projects locally onto the hyperplane w = 0 as a
finite-sheeted cover branched over the zero locus of an analytic function (given
generic (z') the Weierstrass polynomial h(w) = h(z\w) has d distinct roots unless
(z') lie in the discriminant variety of h(w)). Since for most choices of the w-axis
an analytic function will not vanish on it, the above furnishes the basic picture of
an analytic hypersurface.
Let M be a complex manifold. The totality of germs of holomorphic
functions at a point x € M forms a local ring, denoted by 0 , whose unique
maximal ideal is given by
m = {the germs vanishing at x}.
An important property of 0 is that it is a Noetherian unique factorization
domain. This is a consequence of the Weierstrass preparation theorem and
Weierstrass division theorem. (We remark that 0 is the stalk of the sheaf 0
over M at x. See §1 of the next chapter.)
' Let V c M be an analytic variety and x 6 V. Also let I denote the ideal
16

of 0 consisting of all germs of holomorphic functions vanishing along V near x.
Then local defining functions of V at x are a set of functions {fi)'")fj.} defined
near x such that the germs defined by them generate the ideal I . The set of
local functions {f,,"",t} are said to be minimal if the germs of any proper
subset of it do not generate I .
An analytic variety V c M is said to be irreducible if it can not be written
as a nontrivial union of two analytic varieties. V is said to be irreducible at x 6
V if it can not be written as a nontrivial union of two analytic varieties in any
neighborhood of x
Suppose V c M is an analytic hypersurface given near x 6 V by
{f(z) = 0}.
Since 0 is a unique factorization domain the germ f can be written as (uniquely
up to multiplication by a unit in ^ )
f = f, ... f ,
1 m'
where each f. is irreducible. Each f. defines an irreducible hypersurface V. near x,
and we have
(♦) V = V, U ■ ■. U V .
^ ' 1 m
Let V c M be any analytic variety. At x 6 V we have
I = {f 6 ^ : f vanishes along V}.
The analytic Hilbert nullstellensatz states that the radical of I is I itself.
Consequently, I can be written uniquely (and irredundantly) as
X 1 m'
where the p.'s are prime ideals in 0 . The p.'s correspond to local irreducible
varieties and V is written as a finite union of irreducibles locally.
At this point we remind the reader that we often use the same notation to
denote a local function and the germ defined by it.
It is easy to see that the finite unions and finite intersections of analytic
17

varieties are again analytic varieties. Another useful observation is that if
f: M -» N
is a holomorphic map between complex manifolds and if W c N is an analytic
variety, then so is f~ (W) C M. On the other hand, given an analytic variety V
C M its image in N is not necessarily an analytic variety. The proper mapping
theorem says that if f|y is a proper map, then f(V) is an analytic variety.
By a complex submanifold S in M we will mean, unless otherwise specified,
a holomorphically embedded submanifold. We can think of S c M locally as the
zero locus of k independent holomorphic functions, {^i)**')^}) on M. (The
functions f.,« • •,f, are independent if and only if df.A* • 'Adf, f 0.) The integer k
is the codimension of S in M.
A point X of an analytic variety V C M is called a smooth (or nonsingular)
point if V is a complex submanifold near x. Thus a smooth analytic variety is
just a complex submanifold. Let V . c V denote the set of nonsmooth points.
V . is itself an analytic variety, and is nowhere dense in V. An important
result on the singular locus is the following
Proposition. V C M is irreducible if and only if V\V , is connected.
Proof. Let V^[r"y^i} ^ t^s connected components of V\V. . Then the
closure of each V' is an analytic subvariety in M. Thus V can be written
uniquely as the union of irreducible analytic varieties
(**) V = Vj U . ■. U Vj^,
where each V. denotes the closure of V^ The result follows, d
1 1
The decomposition in (**) is the global version of the decomposition given
earlier in (*).
Chow's theorem states that an analytic variety in IP° is an algebraic
variety. In particular, a compact complex submanifold of IP° is algebraic. A
proof of Chow's theorem will be given in the next chapter.
18

Definition. A compact complex manifold is called an algebraic, or projective
manifold if it can be holomorphically embedded in some complex projective space.
§5. Dimension Theory
In this section we will give several equivalent definitions of the dimension
of a variety. But first recall the
HolomorpUc Implicit Mapping Theorem. Suppose we have:
a) complex valued functions fj,'",f are holomorphic in a neighborhood of
the origin in C°;
b) f.(0) = 0 for every i;
c) the Jacobian J(f) = {■^) has a constant rank, k, in a neighborhood of
the origin. Then there exist a subspace of dimension n-k, L c C°, neighborhoods
Uj c L and U. C L"^ about the origin, and a unique holomorphic map
such that in U,xU- c C° the zero locus of (f,,'• •,f ) is given by the graph
Let V = V(p) C C° be an irreducible affine variety. We note that V is
topologically connected. Define the Jacobian of V to be the matrix
Although this matrix has infinitely many rows the Hilbert basis theorem assures
us that the rank is finite. Indeed we could have picked a finite set of generators
for p and considered the resulting finite Jacobian matrix.
Proposition. The Jacobian of an irreducible variety V, denoted by J(V), has a
constant rank (which is also maximal) in V\W, where W is a proper subvariety
of V.
Proof. In order for the rank of J(V) to go down at a point certain minors have
19

to vanish. Setting a minor equal to zero gives a polynomial equation in the f.'s
which in turn gives a polynomial equation in the x*'s. D
A point of V where the rank J(V) is not maximal is a singular point of V.
Thus V\V . is a complex manifold.
* sing ^
Definition A. Let V be an irreducible affine (hence, analytic) variety in C°. The
dimension of V is defined to be the dimension of the complex manifold V\V. .
*^ ' Mng
If W is an arbitrary variety (not necessarily irreducible) in C°, then we define the
dimension of W at a point x 6 W to be the maximum dimension of all
irreducible varieties in W containing x. (This makes sense since W is at most a
finite union of irreducible varieties.)
Given a projective variety V c IP° we can define the dimension at x € V to
be simply the dimension at x of the corresponding homogenous affine variety
minus 1. Equivalently, dim V is the dimension at x of any affine part of V
containing x.
If V is an irreducible (affine or projective) variety, then dim V is clearly
constant. Also the dimension of the empty variety is usually taken to be -tp.
There are two purely algebraic ways of defining the dimension of a variety
which we discuss now. Let V c C° be an irreducible variety and also let p be the
prime ideal in C[x ,• • •,x°] defining V. We then have
Definition B. The dimension of V is the transcendence degree of the extension
CcR^ = C[x\...,x°]/p.
Observe that if R^ is a purely transcendental extension of C of degree d,
then V is isomorphic to C .
By a prime chain of length m at p (p, a prime ideal in C[x ,"«,x°]) we
mean a strictly increasing sequence of prime ideals of the form
PC pj C ••• C p^C C[x\...,x°].
Such a chain corresponds to a strictly decreasing chain of proper subvarieties
20

V(p) D V(Pj) D '" D V(pJ i 0.
We have the well-known
Jordan-H5lder Theorem. Fix a prime ideal p c C[x .•••,x°]. Then any prime
chain at p can be refined to a maximal chain. Furthermore, any two maximal
chains at p have the same length.
Definition C. Let V = V(p) c C° be an irreducible affine variety. Then the
dimension of V is the length of a maximal prime chain at p.
Note that in a maximal chain of length m, p will always be a maximal
ideal, i.e., V(p ) will be a point variety.
m'
To give a simple illustration, we consider
p = <x^>, pj = <x\x^>, .■■, p^_j = <x\...,x°>,
all prime ideals in C[x ,• ■ ^x"]. We then have
V(p) = the x^...x°-plane, V(pj) = the x^• ■ x°-plane, •••, V(p^_j) = {0}.
We see that the rank of the Jacobian of V = V(p) is 1. Also
R^ = C[x\...,x°]/<x^> ^ C[x2,...,x°],
hence
transcendence degree R-./C = n-1.
A maximal prime chain at p is given by
PC pj C ••• C p^_j C C[x\...,x"].
A variety (affine or projective) is called a hypersurface if it has
codimension 1 everywhere. In general a variety whose dimension is everywhere
equal is said to be pure dimensional.
A variety is called a curve if it is of pure dimension 1. So the function
field of a curve is a transcendence degree 1 extension of C. A classification of all
such fields would give a birational classification of curves.
From Definition C we see that an irreducible affine variety defined by a
prime ideal p is a curve if and only if every prime ideal containing p is a
21

maximal ideal. On the other hand we have
Proposition. A variety V c C° (respectively, V c IP°) is a hypersurface if and
only if it can be defined by a single nonconstant polynomial in C[x .•••,x°]
(respectively, a single nonconstant homogneous polynomial in C[x .•••,x°]).
Proof. We first consider the case where V is an irreddudble affine variety in C°.
Suppose V = V(f,,'",f ), f. nonconstant polynomials. Consider f.. Suppose
f =s f ••• f
h 11 la
is a factorization of f. into irredudbles. Then
v(fi) - v(fjj) u. ■. u v(f J.
Hence V C V(fj.) for some i, since V is assumed to be irreducible. Since f.. is
irreducible we then must have V = V(f .). Conversely, suppose that
V = V(f), f a nonconstant polynomial.
The dimension of V(f) is constant since V is assumed to be irreducible. Now for
some i, -x-^ is not identically zero. Hence -^-i can not vanish on V, for otherwise
or
it would have to be in the prime ideal generated by f while deg(^J is strictly
less than deg(f). Thus the rank of J(V) attains the maximum of 1 at a point of
V. Hence the dimension of V is n-1. This takes care of the irreducible affine
case. Now any affine hypersurface is a finite union of irreducible hypersurfaces
and the result generalizes easily. For a projective variety one dehomogenizes and
reduce it to the affine case, d
We know that for any pair of subspaces L-, L^ C C°, or for any pair of
projective subspaces L., L c 1P°
codim(Lj n L ) < codim(Lj) + codim(L ).
For example, any two planes through the origin in C must intersect nontrivially.
In fact, it is not hard to prove the following
Proposition, a) H V , V. are any two varieties in IP°, then
codim(Vj 0 Vj) < codim(Vj) + codim(V2);
22

b) the same dimensional relation holds for affine varieties in C" given that
they intersect each other.
We observe that for a generic pair of varieties
codim(Vj n Vj) = codim(Vj) + codim(V2).
We say that generic varieties intersect properly.
A proof of the following embedding theorem will be given in a later
chapter.
Theorem. Any m-dimensional projective manifold can be holomorphically
embedded in IP^'""'"^
§6. The Degree
In this section we shall restrict our attention to projective varieties.
Let M(n+l,C) denote the set of all (n+l)x(n+l) complex matrices. Also
let GL(n+l,C) C M(n+l,C) denote the set of all nonsigular matrices. The
determinant map
Det: M(n+l,C) -» C
is a polynomial in C[(x!)], where (x!) denote the usual matrix coordinates giving
an idenitification of M(n+l,C) with C^""*" ' . We see that
GL(n+l,C) = c(°+^)\Det-^(0).
M(n+l,C) acts on the lattice of varieties in IP° as follows: Given a
projective variety V in IP° let V, denote the corresponding homogenous affine
variety in C""^ . For A e M(n+l,C), A«V is then the projective variety
corresponding to the homogenous variety given by
{A(v) = Av: V - V,---,x°) e VJ C e-^\
IPG(n,k) denote the Grassmann manifold of projective k-planes in IP°. Then
GL(n+l,C) acts on IPG(n,k) as any k-plane in P" is a variety, and this action is
23

transitive.
Definition. By a generic subset of M(n+l,C) we will mean a subset whose
complement is a proper affine subvariety in M(n+l,C) = c'""*"^ .
Observe that a generic subset of M(n+l,C) is "large". For example, a
generic subset of M(n+l,C) is open and dense in M(n+l,C).
Let V C P° be a hypersurface defined by the product
f = f, ... f
1 m
of a finite set of distinct irreducible homogeneous polynomials, and let d denote
the degree of f. Then there exists a generic subset S C GL(n+l,C) of M(n+l,C)
such that for any A 6 S, L 6 PG(n,l), the intersection
A(L) n V
consists of exactly d distinct points. To put it simply, the intersection between a
degree d hypersurface and a general line consists of d distinct points.
More generally, let V C P° be any pure dimensional variety of dimension
m. Then there exists a generic subset S c GL(n+l,C) of M(n+l,C) such that for
A 6 S, and L 6 PG(n,n-m), A(L) and V meet at a common fixed number of
distinct points. This number is called the degree of V.
Remark. As we shall see later there are several other equivalent ways of defining
the notion of degree. Identify H . (P°,ff) with ff by taking the homology class of
a P as the generator. Then the degree of a k-dimensional variety V in P° is
just the underlying homology class of V in H,. (P°,J). Another useful description
of the degree of V C P° is furnished by Wirtinger's theorem which states that
deg(V) = Vol(V)/k!,
where the volume of V is computed relative to the standard Fubini-Study metric
onP°.
Let v., V- c P° be pure dimensional varieties of dimensions m. and m
respectively with m.+m, > n. We also assume that V. and V, intersect properly
24

excluding the case where they have an irreducible component in common. We
then have the following
Pioposition. There exists a generic subset S C GL(n+l,C) of M(n+l,C) such that
for A., A , A 6 S, L 6 IPG(n,2n-m.-m ), the intersection
Aj(Vj) n A^CVj) n AjCD
consists of a common fixed number of distinct points. This number, denoted by
#(¥^,¥2), is called the intersection multiplicity.
An elementary proof of the above proposition is found in [K] Chapter IV.
We will examine the notion of intersection multiplicity from different points of
view in later chapters.
We now mention the well-known
Bezont Theorem. Let V. and V. be pure dimensional projective varieties
intersecting properly. Then
#(Vi,V2) = deg(Vi).deg(V2).
For example, two curves in IP of respective degrees d. and d. having no
component in common can intersect at most d ^d times.
We close this section vrith a result giving the minimal degree of a pure
dimensional variety in IP°.
Theorem. Let V be an irreducible k-dimensional variety in IP°. Further suppose
that V is nondegenerate, i.e., it does not lie in any hyperplane. Then
deg(V) > (n-k) + 1.
Proof. If the codimension n-k is 1, then the result is trivial. We suppose that
the codimension is not 1. Take a general point p e V and consider the space of
lines in IP° through p. This space can be identified with IP°~ . Consider the
projection given by
T : IP° -» IP°"\ X H L ,
p ' px'
where L denote the line through p and x. This map is holomorphic, and it is
25

not difficult to see that the degree of t (V) in IP°~ is strictly smaller than the
degree of V in IP°. Since the codimension of V decreases by 1 upon applying the
map T , we are done by induction, d
An immediate consequence of the above theorem is that an irreducible
k-dimensional variety of degree d must lie in a IP "*" ~ . In particular, if the
degree of V is 1, then V must be linear.
An irreducible variety V c iP° is called a variety of minimal degree if it is
nondegenerate and
deg(V) = codim(V) + 1.
These varieties were completely classified by Del Pezzo and Bertini. (A smooth
variety of minimal degree is a quadric hypersurface, a rational normal scroll, or
the Veronese surface in IP .) For an excellent account of this material we refer
the reader to the article [Ha2].
§7. Simplidal Homology and the Intersection Number
A g-simplex, denoted by A = A**, in IR° is the convex hull of q+1 points
Vj.,'-',v € IR°, where {v.-Vq, •••, v -Vj.} is a linearly independent set of q
vectors. The points Vq,'",v are called the vertices of A and we write
A = A(vp-..Vq).
A subset {v. , • • • ,v. } c {v., • • • ,v } determines a j-simplex which is called a
j-face of A. A simplidal complex K is a finite collection of simplices in some IR°
such that
a) if A € K then all faces of A are also on K;
b) if A, A 6 K, then either AnA = 0orAnAisa common face.
The subspace
26

|K| ^ U A c R''
is called the underiying space of K. The dimension of K is the maximum of the
dimensions of the simplices of K.
Let X be a topological space. A triangulation of X is a pair (K, tp), where
K is a simplicial complex and
yj: |K| -X
is a homeomorphism. A topological space admitting a triangulation is called a
polyhedron. For example, the two-sphete S is homeomorphic to a tetrahedron
in IR and the "tetrahedron complex" contains 14 simplices.
Let A = A(vp«"v) be a q-simplex and also let (i^'-'i), (L'-'j)
represent two orderings on the vertices Vq,'",v . Then (in'-'i ) and (Jq"*J )
are said to define the same orientation on A if they are related by an even
permutation. So there are exactly two possible orentations on A. A q-simplex
together with a fixed orientation is called an oriented q-fiimplex. By way of
notation we write
A = A[vq...vJ
to denote an oriented q-simplex determined by the ordering (O-'-q). We also
write -A for A[v,VrtV„« • -v ].
Let X be an oriented n-manifold and also let (K, tp) be a triangulation of
X. Then the orientation of X induces via tp an orientation on each n-simplex of
K such that the oriented n-simplices in turn induce coherent orientations on their
(n-1) faces, i.e., whenever an (n-l)-simplex is a face of two n-simplices the two
induced orientations on it are opposite.
Let S denote the set of oriented q-simplices of a simplicial complex K. So
for q > 0, IS I is twice the number of q-simplices in K. Picking orientations we
write, for q > 0,
27

S = S+ U S"
q q q
Note that S^ is simply the set of vertices in K.
Defioition. Let 0 < q < n =s dim K. A q-chain of K is a function
f: S -» ff
q
such that for q > 1, f(-A) = -f(A) for every A e S .
C (K) denotes the set of all q-chains. C (K) is naturally an abelian group:
(f+g)(A) = f(A)+g(A), etc. It is called the q-th chain group of K. Given A in
S"^ define a q-chain
f.: S -» ff
A q
by the prescription
f^(A) = 1; f^(-A) = -1; f^ = 0 otherwise.
It is convenient to identify A with f.. One easily verifies that
C (K) = {m,A.; m. 6 J, A, e S+}
q^ ^ ^111 ' 1 q-*
is the free abelian group generated by the simplices in S"*".
The q-boundary map fl = fl : C (K) -* C ,(K) is given as follows:
a) for 1 < q < dim K, d is the homomorphism determined by
b) for q < 0, and q > dim K, a = 0.
Put
Z^(K) = Ker(fl^), B^(K) = Im(fl^^j).
Elements of Z (K) are called q-cycles and elements of B (K) are called
q-boundaries. A routine combinatorial argument shows
a ofl . = 0
q q+i
so that B (K) c Z (K). The q-th homology group of K is defined to be
H (K) is a finitely generated Abelian group. The rank of H (K) is called
28

the q-th Betti number.
If X is a polyhedron with a triangulation (K, tp), then the q-th simplicial
homology group of X with integer coefficients is by definition
H^(X,J) = H^(K).
It is a theorem that the isomorphism class of this group is a topological invariant
of X.
For the rest of this section we let M denote a smooth oriented compact
n-manifold. Such a manifold is a polyhedron. That is to say, there exists a
simplicial complex K and a homeomorphism, called a triangulation of M,
r. |K| -» M.
In fact we can choose the triangulation to be smooth in the following sense: the
map r restricted to A** c K** can be smoothly extended to a neighborhood of A**
in IR**. Given a fixed smooth triangulation we often identify A** € K with its
image in M and speak of a smooth q-simplex in M. Similarly, we will speak of
smooth chains in M. (A smooth chain in M is a finite integral sum of smooth
simplices in M.)
Given two homology classes
a 6 H^(M) and 0 6 H^_^(M),
we can always find smooth cycles A** and B°"^ in M representing o and 0
respectively. Moreover, the cycles A,B can be chosen such that whenever they
intersect they do so transversely. (Recall that two submanifolds S., S. c M are
said to intersect transversely if at every point x in the intersection
T S, + T S, = T M.)
X 1 X 2 X '
Suppose we have two cycles A**, B°"^ embedded in M, and p e A n B a
point of transverse intersection. Let v,,'",v € T A c T M be an oriented basis
1 q P P
for T A and also let w,,'««,w 6 T B form an oriented basis for T B. We
p 1' n-q p p
then define the intersection index of A with B at p to be
29

+1 if v,,« • '.v ,w,,« • ^w is an oriented basis for T M;
1* * q* 1* ' n—q p '
-1 Otherwise.
If A and B meet transversely everywhere, then we define the intersection number
i(A,B) to be the sum of all intersection indices. The intersection number i(A,B)
depends only on the homology class: if A and B are cycles homologous to A and
B respectively, then i(A,B) = i(A,B). Extending linearly over the integers we
thus obtain a bilinear map
i: H^(M,J) X H^_^(M,J) -. I.
By way of notation
H^(M,(|) = H^(M,ff) .2 i
So H (M,(|) is a /5 -dimensional vector space over (|, where 0 denotes the q-th
Betti number. The intersection pairing extends to
i: H^(M,(|) X H^_^(M,(|) - tl.
Poincare Duality. The pairing i is nondegenerate.
To put it another way, we have an isomorphism
H^(M,<) - (H^^(M,<))* = Hom(H^_^(M,<),<)
given by
a H l^, Ip) = i(a,^).
We will give a proof of Poincarl duality using the Hodge theorem later.
Remark. The Universal Coefficient Theorem implies that the q-th simplicial
cohomology group H**(M,J) is (noncanonically) isomorphic to
F^(M,J) • T^_j(M,ff),
where F (M,lf) denotes the free part of H (M,]7) and T _j(M,lf) is the torsion
subgroup of H (M,ff). On the other hand, the strong form of Poincare duality
states that H*'(M,ff) is canonically isomorphic to H (M,ff). Consequently, we
have in addition to the rank equality 0—0
T^(M,J) ^ T^^_j(M,ff).
30

However, this isomorphism is less useful as it is noncanonical.
§8. De Rham Cohomology and Poincare Duality
Let K be a (finite) simplicial complex. Recall that the q-th chain group
C (K,IR) consists of real linear combinations of oriented q-simplices of K. Put
C''(K,IR) = (C^(K,IR))*.
The Abelian group C''(K,IR) is called the q-th (real) cochain group. We have the
q-th, 0 < q < dim K, coboundary map
^ = tf = d*: ^^{KJR) -* C''+^(K,(R),
*(V)(C) = ¥<fl(c)), c 6 C^^j(K,IR),
where 6: C , j(K,IR) -* C (K,IR) is the (q+l)-th boundary map. The map 6 is
the adjoint of d:
<Sip, c> = <ipy dc>.
Put
Z''(K,IR) = Ker(^), E''(K,IR) = Im(^-^).
Since fl^ = 0 we also have ^ = 0. Hence B''(K,IR) C Z''(K,l!l), and we define the
q-th simplicial real cohomology group to be
H''(K,IR) = Z'^K,IR)/B''(K,IR).
It is a routine exercise to show that the group fl''(K,lR) is (naturally)
sic
isomorphic to (H (K,IR)) : Consider the pairing
H''(K,IR) X H (K,IR) -» R, <f, x> = f(x),
where f 6 Z''(K,IR), x e Z (K,lR) represent f, x respectively. One shows that this
pairing defines an isomorphism
H'^K.R) -. (H^(K,R))*,
f H L^, Lj(x) = <f, x>.
We now let M be a smooth manifold. Let A*'^(M) denote the space of
31

smooth q-forms on M. We put
z2(M) = {ip 6 A^uy. dv? = 0},
BJCM) = {<p 6 A'^CM): if = dip for some V 6 A^^'^M)}.
Elements of 2i^{M) are called closed q-forms, and elements of B2(M) exact
q-forms. Since dod = 0, exact forms are closed. This leads to the q-th de
Rham cohomology group of M
H3(M) = Z3(M)/B2(M).
Note that the de Rham cohomology is a cohomology over the reals.
Let M be a smooth compact oriented n-manifold, and fix a smooth
tri angulation
r: |K| -» M
from a finite simplicial complex K. We identify |K| with M, and put
H''(M,IR) = H'^CK.R).
In the following we will give a homomorphism
HJCM) -» H'^M.R).
First of all observe that to give such a homomorphism it is enough to give a
sequence of homomorphisms
such that
1 = 1: A^^CM) -» C^iUJR) = (C (M,IR))*
fl*oI = I ,,od.
q q+i
Given a smooth q-form ^ on M we define I(^) = I to be the linear functional
C^(M,R)^R, yc)
■!'■
•'c
Remark. If w 6 A*'(M), then the integral
f T*U
makes sense: A is an oriented q-simplex of K and t*(jJ is the pullback to A via
the triangulation map. It is a matter of notation that
32

W = T*tJ.
J A J A
A JA
Extending this definition linearly over the reals we then can integrate any q-form
on M over any smooth q-chain in M. Moreover, if o and 0 are two smooth
q-cycles homologous to each other, then
w = w + drj
'/J
for any u 6 Z^(M), tj e A**" (M): Firstly we have
V
in it
by Stoke's theorem. Bat dP = 0 since /5 is a cycle. Since o and 0 are
homologous there is a (q+l)-chain 7 such that o = 0+8'y. Now
I w = I dw = 0
since uj is closed.
Coming back to our main discussion we note that the sequence (I ) defines
a homomorphism
[■
H3(M) -* H''(M,IR) = (H (M,IR))
♦
The de Rham theorem states that this map is in fact an isomorphism. (For a
proof of de Rham's theorem see [ST] pp. 165-173.)
Any compact oriented topological manifold is a polyhedron, hence we can
construct its simplicial cohomology groups. However, to define the de Rham
cohomology group it is necessary to have a differentiable structure on the
manifold. However, de Rham's theorem tells us that the de Rham cohomology
groups depend only on the topology and not on the particular differentiable
structure used to construct them.
By the remark above we have the integration pairing
H (M,IR) X HJ(M) -. R, ([a],[w]) h f u.
This pairing is in fact nondegenerate, i.e., the linear maps
33

H (M,IR) -* (H3(M))*, [a] » f ,
H3(M) ^ (H^(M,IR))*, [a;] h j^
are isomorphisms. The integration pairing gives explicit identifications
HJ(M) ^ H'^CM.R) ^ (H (M,IR))*.
Poincar^ duality can be restated as follows: the pairing
HJCM) X Hp(M) - R, {[<pm) H [ v,A V
^ M
is nondegenerate. We will give a proof of Poincar6 duality in this form in the
next section.
We thus have natural identifications
H^(M,R) - (HJ(M))* ^ Hp(M),
H2(M) % (H^(M,R))* - H^^(M,R).
In particular, given a smooth q-cycle o in M there exists a closed
(n-q)-form oj such that
V> = v>A w
for every closed q-form (p. Moreover, the cohomology class of oj is uniquely
given. We call the class of u), the Poincar^ dual of o.
A variation on the above theme is given in the following
Pioposition. Let M be a smooth compact oriented n-manifold, and let S c M be
a closed oriented submanifold of dimension p. Then S defines a unique de Rham
cohomology class [<pA 6 H°~''(M) with the property that
for every o e Z^(M).
Proof. The submanifold S defines a linear functional on H^(M) by integration.
Poincar^ duality does the rest, d
The class [<p] will be called the Poincar^ dual of S.
34

§9. Tbo ^odge Theorem
Let M be a smooth oriented compact Riemannian manifold of dimension n.
As in the previous section we let A'^CM) denote the real vector space of smooth
p-forms on M. Note that A (M) is the space of smooth functions on M. Choose
a local orthonormal frame e = (e.) in M, and let 0 = (^) be the dual coframe.
The Hodge operator is the linear map
♦: AP(M) ^ A""P(M), 0 < p < n,
determined by
*{0h"-hf) ^ ±1, *(i) = ±((?^a-..aO, *(^a.-.aO = ±(^"*"^a...aO,
where we take the positive sign if (^) is positively oriented, and the negative
sign otherwise. We al$o define 6: h^{M) -* h^ (M) by
6 = (-l)°(P+^)"'"^odo*; tf s» 0 on functions.
Definition. Let M be a smooth compact oriented Riemannian n-manifold. Then
the Laplace-Beltrami operator A of M is defined to be
A: AP(M) -* AP(M), 0 < p < n,
A = 5od + doS.
We make A''(M) into an inner product space by defining
<o, /9> = I oA*/3.
The graded algebra
ACM) = ®S AP(M)
also becomes an inner product space by decreeing that
AP(M) ± A''(M) for p ^ q.
The Laplace-Beltrami operator, also called the Laplacian, is a self-adjoint
operator on the inner product space A(M), i.e.,
<Ao, p> - <o, A0>, a,0 6 A(M).
This follows from the observation that <do, P> = <o, Sp>.
35

A p-form tj 6 A'*(M) is called a harmonic p-form if Aw = 0. So
tj 6 A'*(M) is harmonic if and only if dw = 6u) = 0.
We can now state the
Hodge Theorem. Let ^^{M) denote the space of harmonic p-forms on a compact
oriented Riemannian manifold M. Then
a) dim TP(M) < oo,
b) AP(M) = A(AP(M)) 9 tP(M), i.e., ACA'^CM)) = t^iU)':
Before proving the Hodge theorem we first recall some preliminary results
from analysis. Suppose we want to find tJ 6 A'*(M) satisfying the equation
(t) Aa; = a,
where o is a given p-form. If w is a solution, then
<Aw, (p> = <o, <p>
for every (p 6 A'*(M). So
<u, ad(A)^> = <Uy A^> = <o, (p>.
Thus u) determines the bounded linear functional on A'*(M)
such that
lJA,p) =: <a, ip>
for every <p 6 A'*(M). A bounded linear functional / on A'*(M) with the property
that
i(A^) = <o, ip> for every (p e A'*(M)
is called a weak solution to (f). We now recall three well-known results from
analysis.
Fact 1. Suppose a bounded linear functional / on A'*(M) is a weak solution of
(t). Then there exists u e A'*(M) such that
l((p) = <oj, <p>
for every (p 6 A^(M), i.e., / = / . Consequently, u solves (f).
w
36

Fact 2. Let (o ) be a sequence of smooth p-forms on N with
la I < c and lAa I < c
' n' - ' n' -
for every n and for some constant c > 0. Then there exists a subsequence of
(o ) which is a Cauchy sequence in A'*(M).
Fact 3 (Hahn-Banach). If S. is a subspace of a normed linear space S, and if f
is a bounded linear functional on S , then f can be extended to a bounded linear
functional F on S^ such that |F| = |f|. (Recall that
|F| = sup |i^: X 6 S^UO}}, etc.)
We now prove the Hodge theorem. Our proof follows [Wa] pp. 223-225
rather closely.
Proof of the Hodge Theorem, To prove a) we suppose that dim if (M) = od.
Then 'X^{M) would contain an infinite orthonormal sequence. By Fact 2 this
orthonormal sequence would contain a Cauchy subsequence, and this is absurd.
To prove b) we let u.,"-,(J, be an orthonormal basis of Tt^iM). Then any o in
A'*(M) can be uniquely written as
o = fl + S <a,(j.><^'-
'^ 11
(This equation defines /5, and clearly P ± u. for every i.) Put
TP(M)-' = {0 6 AP(M): /5 j. 0/., 1 < i < k}.
We will show that TP(M)-' = A(AP(M)). Easily Tt^iU)"- 3 A(AP(M)) for if u is
in A'*(M) and o e Tl^iM), then <Aw, o> = <w, Ao> = 0. We claim that there
exists c > 0 such that
1^1 < c|A^| for every 0 € ^^(M)^.
Suppose the contrary. Then there exists a sequence
(/?.) c IfP(M)^ with |/?.| = 1 and |A/?.| -» 0.
By Fact 2 there exists a subsequence of (/?.) which is Cauchy. Relabeling we
assume (p.) to be Cauchy. So the limit of <0.,(p> as j -► od exists for any <p in
AP(M). Define / € AP(M)* by
37

<V7) = lim </5., ip>.
I is bounded, and
I{A<p) = lim <0., A^> = lim <A0., ip> sj o.
So / is a weak solution to the equation A/5 = 0. By Fact 1 there exists 0 in
AP(M) such that
K<P) = <P, V», <P 6 AP(M).
Consequently 0. -» 0. Since |/5.| = 1 and /5. 6 T^iU)"- it follows that |/5| = 1
J J J
and ^ 6 Tt^iU)^. But A^ = 0, i.e., P 6 1fP(M). The contradiction establishes the
claim. Returning to the main proof now let o 6 7?{M)^. We will show that o
is in A(AP(M)). Define / e A(AP(M))* by
l{A<p) = <o, (p>, <p 6 A(AP(M)).
/ is well-defined: if A^^ = A(p^, then Vj-^g ^ ^'^(M) and <o, <p,-V2> - 0-
Also / is a bounded linear functional on A(AP(M)): To see this let ^ 6 A'*(M)
and also let
^ = ^ - E «p,tj.>tj..
Using the claim we have
\l{A,p)\ = |i(AV)| = \<a,ip>\ < |a||V| <c|a||AV| =c|a||Av>|.
So / is bounded. By the Hahn-Banach theorem /, then, extends to a bounded
linear functional on A^(M). Thus / is a weak solution to the equation Acj = a.
By Fact 1 there exists oj 6 A'*(M) such that Aw = o. Hence o 6 A(A'*(M)). d
Consider the projection
H: AP(M) -» tP(M), o h E<o,w.>w.,
where (w.) is an orthonormal basis for ^^{M). The Hodge theorem implies that
the equation
Aw = a, o e A'*(M) given,
has a solution iff o 6 'K^{M)'-. Given o 6 AP(M)
o - H(o) 6 Tt^iU).
38

Now consider the equation
Aw = o - H(o)
in the unknown a;. If w,)^^ *^® solutions to this equation, then their difference is
a harmonic p-form. Thus there exists a unique solution to the above equation in
1f'*(M)"^. Let G(o) denote this solution so that for any o 6 A'*(M)
o = H(a) + AG(a).
The map
G: AP(M) -» i^iU)"-, OH G(o),
is called Green's operator. Green's operator commutes with d,6, and A.
In what follows we give a proof of Poincar^ duality.
Proposition. Each de Rham cohomology class on a compact oriented Riemannian
manifold M contains a unique harmonic representative.
Proof. Let o e A^{U). Now
o = dotfoG(o) + ^odoG(o) + H(o)
= dotfoG(o) + tfoG(do) + H(o).
Assume o to be closed. Then
o = dotfoG(o) + H(o).
So [o] = [H(o)] 6 HP(M). Now let a .o. be harmonic p-forms with
a^ - a^ = d^ 6 BP(M).
Then
0 = d^ + (Oj-Og), and
<d0, aj-a2> = <0> tfaj-da2> = <^.0> = 0.
So d/3 = 0 and o^ = a^. a
We thus obtain the vector space isomorphism
HP(M) w 1f^(M), [.] M the harmonic representative.
Proof of Poincare Duality. We will show that the pairing
39

HP(M) X h;-P(M) - R, {Wm)» f VAV
is nondegenerate. Let 0 ^ [<p] £ HP(M). We will find 0 ^ [V*] € Hj~P(M) with
([^1>[V'1) f 0. Without loss of generality we assume that (p is harmonic. Since
*oA = Ao*, *^e Z°"''*(M) is also harmonic. Now
Then ij; = *(p does the job. Finally observe that any smooth manifold can be
given a Riemannian metric, hence Poincar^ duality applies to any smooth
oriented compact manifold, d
40

Chapter II. Transcendental Methods
The language of sheaves provides an important technical tool in the study
of complex manifolds: its use results in greater unity and conceptual
simplification. Sheaf cohomology groups via Cech theory are defined in §1. We
also introduce Dolbeault cohomology groups in §1. In §2 we discuss complex
vector bundles, and give Chern's curvature-theoretic formulation of characteristic
classes. Along the way we give an exposition on the theory of metric and type
(i,0) connections on Hermitian vector bundles. In §3 we discuss divisors and
associated holomorphic line bundles on a compact complex manifold; linear
systems of divisors and the corresponding holomorphic maps into projective
spaces. Hodge theory for KShler manifolds (and more generally for Hermitian
vector bundles) is sketched out in §4; we derive various relations amongst Hodge
numbers. In the last section of the chapter we look at some commonly occuring
bundles over projective space: the holomorphic tangent and cotangent bundles,
hyperp'ane bundle, universal bundle, canonical bundle, and so on. Several ways
of calculating their Chern classes are also discussed.
The standard reference on the materials presented in this chapter is [GH].
The book [W], albeit less comprehensive, also gives a very readable exposition on
many of the topics covered here. In particular, a treatment of sheaf cohomology
via abstract soft resolutions can be found there.
§1. Sheaves and Cohomology
Let X be a topological space. A presheaf 7 of Abelian groups on X is
given by two pieces of information:
a) for every open set U C X we are given an Abelian group ^U);
b) for any pair of open sets V c U of X there is a homomorphism, called
41

the restriction map, p„„: ?(U) -* ?(V) such that
^UU "^ identity, p^^ = Pyfy°Py^ whenever W C V C U.
Recall that a directed set A is a set with a partial order, denoted by <,
such that
V a,b 6 A there exists c € A with a < c and b < c.
A direct system of sets indexed by A is a collection {S : a € A} of sets together
with maps
/?, : S -* S, for a < b;
p = id, p = p , OA for a < b < c.
'^aa * '^ca '^cb '^ba
Let {S : a € A} be a direct system, and consider the disjoint union
II S , a € A.
a'
On it we introduce an equivalence relation by decreeing that
X € S^ « y € S^, if 3 c > a,b with p^J,x) = p^^{y).
The above disjoint union modulo the equivalence relation is called the direct limit
of the system and is denoted by 1 i m S .
Suppose we are given a presheaf {?(U): U, open in X} on X. We fix a
point X € X. Then the set
{?{\J): U contains x}
becomes a direct system by setting
U < V whenever V C U:
if X € U n V, then U n V M and U n V > U, V. The stalk of ? at x € X is
defined to be
?^ = Ij^m ?(U), X € U.
A sheaf over X, which is a fibre bundle, gives rise to a presheaf so that the
stalks of the presheaf are the fibres.
Definition. Let X be a topological space. A sheaf of Abelian groups over X is a
topological space S together with a map t: 5 -* X such that
42

a) T is a local homeomorphism;
b) for every x € X, t~ (x) = <S is an Abelian group;
c) the group operations are continuous, where on S if S we use the product
topology.
A section of S over an open set U C X is defined to be a continuous map
f: U -* «S with Tof = id.
The set of all sections over U will be denoted by «S(U). The presheaf
{5(U): U, open in X}
is called the presheaf associated to S. It is not difficult to verify that the stalk
at X e X of the associated presheaf is exactly tT (x) = S .
Examples
a) Let G be an Abelian group endowed with the discrete topology. Then
the constant sheaf, denoted again by G, on X is given by
r. G X X -* X, (g,x) w X.
Note that G(U) = {constant maps} = G, and p^^ = id^ for any open V c U.
b) Let X = M be a smooth manifold. We have
C" -» M,
the sheaf of germs of smooth functions on M. C"(U) consists of smooth functions
on U. The stalk at x € M is the set of germs of smooth functions defined in a
neighborhood of x. We also have
A^ ^ M,
the sheaf of germs of smooth p-forms on M. Also
ff -» M
denotes the constant sheaf of integers.
c) Let M be a complex manifold. Then
denotes the sheaf of germs of holomorphic functions on M. Also
43

(f -» M
denotes the multiplicative sheaf of germs of nowhere zero holomorphic functions
on M. We also have
M-^M, / -» M,
denoting the sheaf of germs of meromorphic functions and the multiplicative sheaf
of germs of not identically zero meromorphic functions on M.
Let A, B be sheaves over X. A continuous map between them preserving
stalks is called a sheaf homomorphism if it induces group homomorphisms on the
stalks. For example, on any complex manifold we have the following exact
sequence of sheaf homomorphisms, called the exponential sheaf sequence:
0-»2^(7-t(7*-»0,
where e(f) = exp(27rif).
Let M be a paracompact Hausdorff space, e.g., a manifold. We take a
locally finite open cover U = {U.} of M. The nerve N(//) of the cover U is an
abstract simplicial complex whose vertices are the members U. of the cover U
such that
A = U. U. • • • U. is a q-simplex iff n U. i 0.
q 10 M ^ U
The intersection of the vertices of A is called the support of A . Fix a sheaf
r. S ^M.
A q-cochain of N(//) with coefficients in the sheaf 5 is a function f which
associates to each q-simplex A 6 N(//) a section
f(Aq) 6 5(support(A^)).
The set of all q-cochains forms an Abelian group C** = C**(N(//),«S). It is the free
Abelian group on the simplices of N(//).
There is the coboundary operator
given by the following prescription: if f 6 C** and A = ^n"'^ +i '^®" ^ ^"
44

C**"*"^ has for A the value
q+l
(^f)(A) = E (-1)' p^ f(U,.. •U._jU .. -U ),
1=0
where p, denotes the restriction to U. n ••• n U ,,.
'^A 0 q+l
We have ^ = 0, and put
Z\lf{U),S) = Ker(^), B\lf{U),S) = Im(^-^), b\n{U),S) = 0.
The q-th cohomology group of the nerve N(//) with the coefficient sheaf S
is defined to be
n'^muM = z'^muM/B'^muis).
Observe that H (N(//),«S) consists of the global sections of «S -* M.
Let il denote the collection of all locally finite open covers of M. Introduce
a partial order on 11 by decreeing that
// < V, if V is a refinement of U.
This makes 11 into a directed set.
Lemma. Suppose // < V, where //, V € IL Choose a function i: V -* // with the
property that i(U) 3 U for every open set U € V. Then the induced map N(V) -♦
N(//) is simplicial. Moreover, i induces a homomorphism
i*: E\lf{U),S) -» H*l(N(V),5),
and this map does not depend on the choice of i as long as i(U) D U, U € V.
Proof. Uq---U € N(Z^) implies that
MU^n ... nu^ci(UQ)n ... n i(u^).
So i(UJ...i(U ) is a simplex. The rest is standard. □
All this leads to
Definition. The q-th cohomology group of M with the coefficient sheaf S is
n\M,S) = Ij^m H*»(N(Z^),5), Ue!d.
We can describe H**(M,«S) as follows. Consider the disjoint union
U H*»(N(Z^),5), U 6 IL
45

For a € H*»(N(Z^),5) and b € H*»(N(V),5), a ~ b if and only if there exists HI € il
with iniU.y and i*(a) = i*(b) € H*»(N(V),5). Now
H*»(M,5) = II H*»(N(Z^),5)/~.
Remark, a) For any locally finite open cover //, H (N(//), S) consists of global
sections of «S -* M. So
H°(M, S) = r(M, S).
b) For a sufficiently fine cover //,
H*»(N(Z^), S) = H*»(M, S).
To be more precise, we have the
Leray Theorem. Suppose
H**(support(A ), «S) = 0, q > 0, any p-simplex A .
Then H*(N(Z^), S) % H*(M, S).
For a proof see [GH] p. 40.
Let S, T be sheaves over M. Then a sheaf homomorphism S -* 7 induces
maps
S{\J) -» r(U) for open U C M,
and homomorphisms
C\li{U),S) -» C*»(N(Z^),r), U € ii
And they in turn induce homomorphisms H**(N(//),«S) -* H**(N(//),7'), and
YL\U,S) -» H*»(M,r).
Given a short exact sequence of sheaves over M
there arises a long exact sequence
0 -» H°(M,.4) -» H°(M,B) -» H°(M,C) i H^(M,.4) -» H^(M,B) -»•••,
where tf = 5*: H**(M,C) -* H**"*" (M,.4) are the connecting homomorphisms.
A partition of unity of the sheaf S -* M subordinate to a locally finite
cover U = {U.} is a collection of sheaf homomorphisms
46

T].: S ^ S
'i
such that each r/. vanishes in an open neighborhood of M\U., and E r/. = 1. A
sheaf «S -» M is called a fine sheaf if it admits a partition of unity subordinate to
any locally finite open cover. Examples of fine sheaves are ^'*'**(M), the sheaf of
germs of smooth type (p,q) forms on a complex manifold M. We have:
for a fine sheaf 5 - M, H**(M,«S) = 0, q > 1.
For a proof of this see [GH] p. 42.
As a motivational example we now explain so called the Mittag-Leffler
problem: Given a discrete set of points {z } in a Riemann surface S and a
principal part at each z , does there exist a meromorphic function f on S
n
holomorphic outside {z } whose principal part at z is the prescribed one? This
problem can be given a sheaf-theoretic solution as follows. Take an open cover U
= {U.} of S so that each U. contains at most one point of {z }, and let f. be a
meromorphic function on U, solving the problem in U.. Set
f.. = f.-f. € 0{\J. n U.).
In U. n U. n U. we have fj-+f-K+^- = 0. Solving the problem globally is
equivalent to finding {g. 6 ^(U.)} with
f.. = g.-g. in U. n U..
Given such g.'s, f = f.+g. is a globally defined function satisfying the conditions,
and vice versa. Now
Z\m).t>) = {(fy): Ij+fji+l^i = 0}.
b'(N(»),0) = {(f..); t, = g.-g. for some (g.), g. € OiVfi.
It follows that H (N(//),^) is the obstruction to the Mittag-Leffler problem.
More precisely, consider the following short exact sequence of sheaves over M
0 ^ 0 2, M i M/0 ^ 0.
The data of the above problem consists of a global section of the quotient sheaf
M/0 -* M, i.e., an element g 6 H {M,M/0). The question is whether or not
47

g = j*f for some global meromorphic function f 6 H {M,M).
Recall the long exact sequence
0 -» H°(M,(7) -» H°(M,iO i n\M,M/0) -» n\M,0) -»•••.
We see that g = j*f for some f € E^MyM) if and only if ^g = 0 in E^{M,0).
The Poincaie Dual of an Analytic Subvariety
Let M be a compact complex manifold of (complex-) dimension m. We
give M the orientation induced from its complex structure. If S c M be a
k-dimensional analytic subvariety, then for any o 6 A (M) we have
f da = f a = 0
'S JdS
since the boundary of an analytic variety has real codimension at least two. This
enables us to define a linear functional on H. (M) as before (Chapter I, §8), and
'd
we obtain a unique cohomology class [(pJ € Hi (M) with the property that
j 0 = j 0Mp^ for any 0 6 zf{yL).
The class of ip^ € zj"" ^'^(M) is called the Poincar6 dual of S.
Dolbeault Cohomology
Let M be a complex manifold. Its complexified cotangent bundle
decomposes as
tJm = T*M • C = T^'°M ® T°'^M:
if (z*) are local holomorphic coordinates, then the sections of T ' M -* M are
given locally by
a.(z)dz\
where each a.(z) is a smooth complex-valued functions. We will just write T*M
instead of T*M when there is no real dangei of confusion. There is the
complexified exterior bundle over M given by
48

A(T*M) = eE A*(T*M).
The above decomposition of T*M induces the decomposition
A*(T*M) = e AP'*»(T*M).
p+q=a
Local sections of A'*'**(T*M) are given by
a, . (z) dz'^A • • • A dz'PA dz^^A • • • A dz\
il««Mpji'«'jq^ '
where the a. . . . 's are smooth complex-valued functions. We will use the
il'-Mpji-•-jq
multiindices
The space of sections of A*(T*M) -* M will be denoted by A*(M), or just
A*(M) if no confusion is likely. Also the space of sections of AP'*»(T*M) -» M
will be denoted by A'''**(M). Using the sheaf notation we have
AP'*»(M) = H°(M,^P'*»).
An element of A'*''^(M) is called a type (p,q) form. A type (p,0) form is
called a holomorphic form if it is locally given as a.dz with the a.'s holomorphic.
For any y? 6 A*(M) we have the decomposition
<p= E /'*», /'*» € AP'*»(M).
p+q=a
We have the exterior derivative
d: AP'*»(M) -» AP+^'*»(M) • AP'*»+^(M).
Put
d = a + a,
where
8: AP'*»(M) -» AP+^'*»(M),
~d: AP'*»(M) -» AP'*»+^(M).
So dtp = (dv7)P+^'*» and ~d(p = {dipf''^'^^. Locally, if
V? = a., dz A dz ,
then
49

dip ~ (flajj/8z') dz' A dz^ A dz*",
~d<p = (flajj/flzJ) dzJ A dz^ A dz^.
We put
ZP'*»(M) = {<p € AP'*»(M): lip = 0},
~d
It is routinely verified that (d^ = 0 iff «^ = fl^ = 0 and dd = -Id.)
V- = 0; fl(AP'*»(M)) c ZP'^^+^CM).
All this leads to the (p,q)-th Dolbeault cohomology group
HP'*»(M) = ZP'*»(M)/BP'*»(M).
A variation on the Poincare lemma gives
nl'^D) = 0, q > 1,
d
for D = {(z*) 6 C°: |z*| < c}, a complex polydisc.
We have the
Dolbeault Isomorphism Theorem. Let M be a complex manifold. Then
HP'*»(M) !v H*»(M,nP),
d
where fl^ is the sheaf of germs of holomorphic p-forms on M.
Proof. By the 5-Poincare lemma the following sequences of sheaves are all exact:
0 -» fiP -» ^P'° -» iP'^ -» 0.
~d
0 -» iP'<» -» XP'*» -» iP'*»+^ -» 0,
where .^P'** denotes the sheaf of germs of smooth type (p,q) forms, Z^''^^ denotes
d
the sheaf of germs of 5-closed type (p,q+l) forms, and
h .4P'*» -» iP'*»+^
~d
Since each ^P'** is a fine sheaf we have
tf(M,^P'*») = 0, for i > 0.
50

Thus the long exact cohomology sequences associated to the above sheaf
sequences yield
d ~ ~d
§2. Complex Vectoi Bundles and the Chem Class
Let t: E -♦ M be a surjective smooth map of smooth manifolds whose fibre
T~ (x) at every point of M is a complex vector space of dimension r.
Definition. We say that E -♦ M is a complex vector bundle of rank r if there
exists an open cover // = {U } of M and fibre preserving diffeomorphsms
ip:B= t"\u ) -» U X C'
which are linear isomorphisms on fibres. (//, ip ) are called trivializations.
Remark. Let (//, <p ) be trivializations of E -♦ M and also let V = {V } be an
open cover of M which is a refinement of U. Then (V, (p \y,) also give
& » ft
trivializations of E -« M.
The maps
are linear automorphisms on fibres, hence give rise to smooth maps
gab= U, n U^ - GL(r,C), g^,(x) = f^o^l'l^.-^^c--
{g ,} are called transition functions and satisfy so called the cocycle conditions
(t)
gab°gbc = Sac ^" u^ f^ u^ n u^.
Suppose we have two sets of trivializations (//, (p ), (//, <p'). Let {g ,},
{g' } be their respective transition functions. Then there exist smooth maps
A^: U^ - GL(r,C)
51

such that
(t) sib = K-i^-^l' '» u. n u,.
Definition. Let M be a smooth manifold, and // = {U } an open cover. Then
any collection of smooth maps
gab'- U, n U, ^ GL(r;C)
satisfying (f) is called a cocycle relative to U. Two cocycles {g , }, {g'u} are
said to be equivalent if they are related by (J).
Given two vector bundles E, E' -* M, a fibre preserving smooth map from
E to E' is called a bundle homomorphism if it is linear on fibres. A bundle
isomorphism E -* E' is a fibre preserving diffeomorphism such that on fibres it
induces linear isomorphisms. It is routinely verified that: two vector bundles
over M are isomorphic if and only if there exists an open cover of M relative to
which their cocycles given by transition functions are equivalent.
Hereafter we do not distinguish isomorphic vector bundles.
Given a cocycle {g , } on M we can construct, unique up to isomorphism,
E -♦ M whose transition functions relative to U are {g ,} as follows:
E = U(U X C) I ~,
where (x,y) € U^ x C ~ (x, gj:x)'y) € U^ x C'.
A complex rank 1 vector bundle over a smooth manifold M is called a
complex line bundle. If M is a complex manifold and if L -* M is a complex line
bundle admitting trivializations (//, tp ) with holomorphic transition functions
^6ab" ^a " ^b ^ G^(l'^) = ^*>'
then L -« M is called a holomorphic line bundle.
Let M be a smooth manifold and also let jt denote the multiplicative sheaf
of germs of nowhere zero complex-valued smooth functions on M.
Proposition. The collection of (isomorphism classes of) complex line bundles over
M is canonically identified with H (M,.4*).
52

Proof. A complex line bundle L -* M corresponds to an equivalence class of
cocycles. Let {g , } be a cocycle representing L -* M. So
where // = {U } is a (locally finite) open cover of M. We see that the collection
{g^j^} gives a 1-cochain of the nerve N(//) with coefficients in jt. {g ,} satisfies
the cocycle conditions (f) which says that {g ,} € Z (N(//),.4*). Another cocycle
{g^j^} represents L -* M iff it is related to {g^j^} by
a' ~ i&.a
^ab fb ^ab
for some f^ € H°(U^,/), fj^ € H°(U^,/) in U^ n U^. (See {%).) This is so iff
It follows that L -♦ M corresponds to a cohomology class in H (N(^),.4*).
Considering trivializations over sufficiently fine refinements we see that L -» M in
fact defines a cohomology class in H (M,.4*). d
Assume that M is a complex manifold. Then using 0* in place of ^ the
above proof yields
Proposition. The collection of all holomorphic line bundles over a complex
manifold is naturally identified with H (M,0*).
Consider the exponential sheaf sequence over a real manifold M
0-»ji.4*/-»0,
where e(f) = exp(25rtf). The associated long exact sequence is
... -» H^(M,.4) -» h\m,/) i h2(M,2) -* H^(M,.4) -»••..
Since .4 is a fine sheaf we get H^(M,.4) = H^(M,.4) = 0. Thus
H^(M,/) ~ H^(M,J).
For a complex line bundle L € H (M,.4*),
Cj(L) = ^(L) e H^(M,J)
is called its Chern class. Thus a complex line bundle over a real manifold is
determined by its Chern class.
53

Assume now that M is a complex manifold. Again we have over M the
exact sequences
... -» Ii\M,0) -» h\m,(?*) -» h2(M,J) -» e\m,0) -»....
The Chern class of a holomorphic line bundle L € H (M,C?*) is
Cj(L) = ^(L) € h2(M,2).
However, 0* is not a fine sheaf.
If f: M -* N is a holomorphic map between complex manifolds and if L -*
N is any holomorphic line bundle, then
Cj(r^L) = f*Cj(L).
Coming back to the general discussion we let E, F -* M be complex vector
bundles of ranks r and s respectively. Generally speaking, operations on the
vector spaces E , F induce operations on the bundles E, F. We give some
examples below.
a) The dual bundle E* -* M is the rank r complex vector bundle with fibre
Transition functions are { g~ }, where {g ,} are the transition functions of E
over M. (Here *g;J(x) = \gJx))-\ x 6 M.)
b) Let (g ,), (h ,) be transition functions of E and F respectively. Then
the bundle E ® F is determined by transition functions {g , ® h ,}.
c) E • F is the rank rs bundle given by {g , • h ,}.
d) A'^E is given by transition functions (A™g ,). In particular, a'E is a
line bundle whose transition functions are given by {det(g ,)}.
If f: M -* N is a smooth map and if E -* N is a complex vector bundle of
rank r, then the puUback bundle f~ E -* M is also a complex vector bundle of
rank r whose fibres are given by
54

Transition functions of the pullback bundle are the pullbacks of transition
functions of the original bundle.
Let M be a smooth real manifold. We consider r. E -* M, a smooth
complex vector bundle of rank r over M. r(E) denotes the space of smooth
sections of E -* M.
Defimtion. A connection on E is a map
V: r(E) -» r(T*M • E),
where r(T*M • E) denotes the space of vector-valued smooth 1-forms on M,
satisfying
a) V(t7j + T/g) = Vr/j + Vr/^,
b) V(fT7) = df.T7+ f Vt7,
where r/., r/ € r(E) and f is a complex-valued smooth function on M.
Fix a complex vector bundle r. E' -► M. A collection of sections
e = (e^.-'-.e^)
over U C M is called a frame over U if at every x e U
ej(x),...,e^(x) span E^ = 5r"^(x).
The connection matrix u = {u)\) of (E, V) relative to e is determined by
Ve. = e. • w? (1 < i,j < r),
or, Ve = eu)y using the matrix notation.
So each a/, is a local 1-form on M.
J
The connection matrix u) determines V. To see this let r/ be an arbitrary
section of E -► M over U.
Tf ~ rfe. for some local functions (rf).
Vr; then must be equal to
V(Vej) = dr/'e. + ri^J^ = {dr^ + rfJ^e..
A section r/ is said to be horizontal if Vr/ = 0. Now Vr/ = 0 if and only if,
in an arbitrary open set U C M,
55

drl + o/'.t' = 0, 1 < i < r.
This leads to an overdetermined system of partial differential equations.
However, restricting to a real curve in M one obtains a system of ordinary
differential equations. Therefore given a curve C C M there exists locally a
unique horizontal section along C.
Let e, e be two local frames. Then, on their common domain, they are
related by
(t) e = eg,
for some g, a GL(r,C)-valued local function on M.
To avoid any possibility of confusion we remind the reader our notational
conventions concerning vectors and 1-forms. We always write a vector
columnwise and a form, rowwise. So
t/..l
We also write
e = (e^,- • ^e ), a basis of C°,
(p zn {(f ," ',(p^), a cobasis.
Proposition. Suppose e, e are related as in (f). Let u (respectively, u) denote
the connection matrix of (E, V) relative to e (respectively, g). Then
&= g~Mg + g~^wg.
Proof. Have Ve = ew, Ve = eu. On the other hand
Ve = V(eg) = edg + ewg = eg~^ (dg +wg) = e(g~^dg + g~^wg). □
The matrix-valued 2-form
fl = dw + w A w
is called the curvature form (or the curvature matrix) of (E, V) relative to e.
Proposition. The transformation rule for the curvature matrix is given by
n = g-^ng.
56

Proof. Have
fl = da; + wA &
= d(g~Mg + g~^wg) + (g~^dg + g~^wg)A(g~^dg + g~^wg)
= g~'ng
using dg ^ = -g Mgg ^ . D
Let E -* M be a complex vector bundle over a smooth manifold M. A
Hermitian structure on E is a smooth field of Hermitian inner products in the
fibres of E. E -» M with a Hermitian structure is called a Hermitian bundle.
Let E -« M be a rank r Hermitian bundle and let <,> denote its
Hermitian structure. Given a local frame e = (e.) we put
h.. = h! = <e., e.>.
Then h.. = fi.., i.e., h is a Hermitian matrix. If e = e-g is another local frame
then, on their common domain, h and h are related by
h = *ghg.
Remark. Our notations are consistent with the following convention: An nxn
Hermitian matrix X defines a Hermitian inner product on C° by the formula
<v, w> — vXw.
Definition. Let E -» M be a Hermitian bundle. A connection V on E is called a
metric connection if
d<C, V> = <VC, V> + <C, Vt7>
for any local sections (, r/.
Proposition. Let E -» M be a Hermitian bundle equipped with a metric
connection V. Fix a local frame e = (e.) and write h, w, fl relative to it. Then
a) dh = *a> h + h a»,
b) 0 = *fl h + h n.
Proof. By definition dh! = d<e.,e.>. Now
d<e.,e.> = <Ve.,e.> + <e.,Ve.> = <e,a;v,e.> + <e.,e, a;v>
57

This proves a). To prove b) we exterior differentiate both sides of the equation
in a) and obtain
0 = (d*w)h + *wA dh + dhA w + h dw = *nh + h n,
since du = -u S u + Q. a
If h is the identity matrix, then a) and b) become
*w + w = 0, *n + n = 0.
That is to say, u) and 0 are skew-Hermitian relative to a unitary frame.
We now suppose that M is a complex manifold, and E -» M is a
holomorphic vector bundle. A local frame s = (s.) of E -* M is called a
holomorphic frame if each s. is a holomorphic section.
Definition. A connection V on a holomorphic vector bundle E -* M over a
complex manifold M is said to be of type (1,0) if its connection matrix relative
to any holomorphic frame is of type (1,0).
Since a; = g~ dg+g~ ug and dg is of type (1,0) for holomorphic g, if the
connection matrix is of type (1,0) relative to a single holomorphic frame then it
is of type (1,0) relative to all holomorphic frames.
Theorem. Let E -* M be a Hermitian holomorphic vector bundle over a complex
manifold M. Then there exists a unique type (1,0) metric connection on it.
Proof. Let s = (s.) be a holomorphic frame and put h'. = <s.,s.>. (Each hi is a
smooth local function on M.) Suppose there exists a type (1,0) metric connection
V on E. Then dh = ^wh+hw by the preceding proposition. Since w is of type
(1,0) we must have *wh 6 type (1,0) and hu € type (0,1). So
8h =■ (Jti and dh = ha;,
where d = 8 + 8, the type decomposition. We see that
(t) OJ = £"^ 8fi (or *w = 8h h"^)
58

is the unique solution to both equations. Since the connection matrix is
determined by the conditions of compatibility (f) defines it globally, d
Hereafter given a Hermitian holomorphic vector bundle we shall use the
connection given by (f) unless otherwise specified.
Upon exterior differentiation the equation in (f) yields
n = - fi~^flfi A fi~^flh + h~HdL
This shows that the curvature matrix 0 is of type (1,1) relative to a
holomorphic frame. But Cl = g~ (Ig and a change of frame does not affect the
type of fl. We thus have: the curvature matrix of the canonical connection on a
Hermitian holomorphic vector bundle is of type (1,1) with respect to any smooth
frame.
Let M be a Hermitian complex manifold of dimension n. This means that
there is a Hermitian structure on the holomorphic tangent bundle TM. Thus TM
is a Hermitian holomorphic vector bundle of rank n. Let e = (e.) be a smooth
frame over an open set U C M. The dual coframe <p ~ (^') is defined by the
requirement v>'(e.) = tfl, 1 < i,j < n, so that
e = eg iff v? = g v>
where g is a local GL(n,C)-valued function on M.
Definition. Consider TM -* M, where M is a Hermitian complex manifold. Fix
a smooth local frame e = (e.) and define ^, u relative to it. Then the torsion
matrix of M relative to e is defined to be
r = d^ + w A ^.
Exterior differentiate both sides of the equation (p ~ g(p and obtain
dv? = dg A v> + g d^ = (g a» - wg ) A g~V + g ^^^
We thus obtain the transformation rule
r = gf.
In particular, the type of r is well-defined independent of the local frame chosen
59

to express it.
Proposition. Let M be a Hermitian complex manifold. An arbitrary connection
V on the holomorphic tangent bundle TM -* M is of type (1,0) if and only if its
torsion matrix is of type (2,0).
Proof. Let s = (s.) be a holomorphic frame, and (p the dual coframe. Write
(jj ~ (jj. + W-, with (jj. 6 type (1,0) and u) 6 type (0,1).
Now T ~ diifi + uS (fi and dy? 6 type (2,0) since s is holomorphic. So
r 6 type (2,0) iff wA y? 6 type (2,0) iff w^A y? 6 type (2,0).
But u S (f £ type (2,0) if and only if w = 0. D
Let M be a Hermitian complex manifold. The Kahler form of M is defined
to be -(j) times the imaginary part of the Hermitian metric. (The real part of
the Hermitian metric is a Riemannian metric making M into a Riemannian
manifold.) If y? = (y?*) is a unitary coframe then the Hermitian metric is given
by
E V?' • (p\
and the Kahler form is given by
M is called a KS.hler manifold if its Kahler form is closed. For example, a
Hermitian Riemann surface is trivially Kahler. We leave the proof of the
following proposition as an exercise.
Proposition. A Hermitian complex manifold M is Kahler if and only if the
torsion matrix of TM -* M (relative to the metric (1,0) connection) vanishes.
Let Mj be a complex manifold and also let M. be a Hermitian complex
manifold. Then a holomorphic immersion f: M. -* M induces a metric on M.
by setting, for v,w € T M.
<v, w> = <f^v, f^w>.
The puUback of the Kahler form of M, is the K&hler form of the induced metric
60

on M .
The Chern Classes of a Complex Vector Bundle
Let M be a smooth manifold, and consider a complex vector bundle E -» M
of rank r. Pick an arbtrary connection V on E, and let (fl..)) 1 1 ij < f) denote
the curvature forms written relative to some local frame. The k-th, 0 < k < r,
Chern class of E is, by definition,
S(E) = [P^2i ")] e Hf (M), Co(E) = 1 6 hJ(M),
where P (•) denotes the k-th elementary symmetric polynomial in the
eigenvalues of the matrix (•). In other words.
detc^ifl + ti)^ E p'>-\-in).tK
^^ k=o ^^
For example.
Ci(E) = [ji trace(fl)].
S(E) = [{-^y det(fl)].
In the above the wedge product is customarily omitted since the fl..'s are all
2-forms and the wedge product is commutative on even degree forms.
k i
One verifies that the form P (— fl) is a well-defined global closed 2k-form
on M. Moreover, the cohomology class defined by it, namely the k-th Chern
class, is independent of the connection chosen on E. For a proof of this, see [C]
pp. 40-43.
The total Chern class of E is defined to be
c(E) = E c.(E) 6 Hf(M).
Suppose L -► M is a line bundle. Then the present definition of the first
Chern class of L coincides with the earlier definition of c.(L) as
tf*(L), tf*.- H^(M,^*) (or h\m,(?*)) -» VL\u,I\
61

once we include H^(M,J) c H^CM) ^ H^(M,IR).
Some of the basic properties of Chern classes are as follows.
a) Let f: M -* N be a smooth map between smooth manifolds, and also let
E be a complex vector bundle over N. Then
c.(r^E) = f*oc.(E).
b) Let E -* M be a complex vector bundle with a connection V and the
curvature form (I. Then the curvature form of E* is -fl. Hence
c.(E*) = (-l)'c.(E).
c) Let E, F -* M be complex vector bundles. Then the Whitney product
formula states that
c(E e F) = c(E).c(F).
Suppose we have an exact sequence of complex vector bundles
0-»A-»B-»C-»0.
Then B is smoothly isomorphic to A ® C, and the Whintey formula can be
applied: c(B) = c(A)-c(C).
d) Let E -* M be a complex vector bundle of rank r, and L -* M a line
bundle. Then
Cj(E . L) = Cj(E) + r.Cj(L).
In the folllowing we will give a proof of d). Pick Hermitian metrics on E
and L. The induced metric on E • L is given by
<a»/, a'»/'> = <a»a'>j,-</»r>j^.
Let V„ be a metric connection on E, V^ a metric connection on L, and V,
metric connection on E»L. Then
Hence
"e.l = V1 + ^»"l'
and
62

Cj(E.L) = [traceC^i n^.^^)] = Cj(E) + r.Cj(L).
Coherent Sheaves
The sheaf of germs of holomorphic sections of a holomorphic rank r vector
bundle over a complex manifold M is locally free, i.e., locally isomorphic to Ol..
Conversely every locally free sheaf of ^^.-modules is the sheaf of holomorphic
sections of a holomorphic vector bundle which is uniqudy determined up to
isomorphism. These sheaves are basic examples of coherent sheaves. A sheaf of
Oj.-modules, S, is said to be coherent if locally there exists an exact sequence of
sheaves of ^^.-modules
"m - "A - -^ - «•
A theorem of Cartan-Serre states that for a coherent sheaf S over a compact
complex manifold M (or more generally over a compact complex space)
dim E\M,S) < m; E\M,S) = 0 if i > dim M.
§3. Line Bundles, Divisors, and Linear Systems
By a line bundle over a complex manifold we will mean a holomorphic line
bundle unless otherwise stipulated.
As we saw in the preceding section a line bundle L -» M may be (and will
be) thought of as an element of the cohomology group H {M,0*). Under this
identification the group operations in H {M.,0*) are given by
L + L' = L •L', -L = L*,
where L* -* M denotes the dual bundle.
Given a line bundle L -* M choose a connection on L, and let Q denote its
curvature form. (Recall that fl is a globally defined closed 2-form on M.) Then
cfi) = Iji ni e h2(m).
63

When the dimension of M is 1 we define the degree of L to be
deg(L) = Cj(L)[M] = 5! f n 6 J,
"'M
where [M] € H (M,J) is given by the natural orientation.
A divisor D on a compact complex manifold M is a finite integral sum
D = E a.v., a. € J,
1 r 1 '
where the V.'s are irreducible analytic hypersurfaces of M. Under addition the
set of all divisors on M, denoted by Div(M), forms a free Abelian group. A
nonzero divisor D is said to be effective or integral (write D > 0) if a. > 0 for
every i.
Let V be an irreducible analytic hypersurface of M. Then a local defining
Junction f at x 6 V is an element of 0 vanishing along V with the property that
if another germ in 0 vanishes along V then it is a multiple of f in 0 . For any
holomorphic function g defined near x 6 V we define the order of g along V at x,
denoted by ord„ (g), to be the largest integer a such that
g = f*'h for some h £ 0 .
Given a holomorphic function g defined on all of M we have
°'^'^V,x(s) = ord^ y(g) for any x,y € V.
This is so since relatively prime elements in 0 stay relatively prime in nearby
local rings 0 and V is connected. We define the order of a holomorphic function
g on M along an irreducible analytic hypersurface V to be ord^ (g) for any x 6
V and write ord„(g).
Let f € H (M,A*), i.e., f is a not identically zero meromorphic function on
M. In a neighborhood of any x 6 M we can find holomorphic functions g and h
relatively prime in 0 with
f = g/h.
Given an irreducible analytic hypersurface V C M we put
64

ord^(f) = ord^(g) - ord^(h).
The divisor of f is defined to be
(f) = E ord^(f).V,
where the sum is taken over all irreducible analytic hypersurfaces V c M. This
sum is finite since the zero set and the polar set of f are both finite unions of
irreducible hypersurfaces. More precisely, let V denote the zero set of g and
also let V denote the zero set of h, both defined locally. Write
V =V U---UV , V. =V. U-.-UV. ,
g gi gj' h hi hk'
where V , V, are locally defined irreducible analytic hypersurfaces. Put
(f)j, = E ord^ (g).V (f)^ = E ord^ (h).V .
gi ^ hi
Then the right hand sides patch up to give globally defined divisors {f)^ and (f) .
We call (f)jj the zero divisor of f, and (f) the polar divisor of f. Have
A divisor D on M is called a principal divisor if there exists a meromorphic
function f with (f) = D.
Proposition. Let M be a compact complex manifold. Then
a) Div(M) is naturally isomorphic to H (M, jt/0*);
b) there is a canonical homomorphism
Div(M) -» li\M,0*), D M Lp,
whose kernel consists of the normal subgroup of principal divisors.
Proof. Given D = E a.V. e Div(M) we can find a locally finite (in fact, finite)
open cover // = {U } of M such that in each U every V. has a local defining
function g. 6 H°(U ,0). Set
1& &
t. = n g«^ e H»(U,,/).
1
It is easy to see that in U^ n U,
i e H'lU. n U,, 0*).
65

Thus {f } defines an element in H°(M,//(?*). We write (f ) € li^{UX/0*) to
denote this element, {f } are called local defining functions for D. Conversely, a
global section of jt/0* -» M is given by an open cover // = {U } of M and
meromorphic functions s ^ 0 in U with |* 6 H (U n U , 0*). So for any
irreducible analytic hypersurface V c M we have ordy(s ) = orfi„(s,). Hence we
can associate to s the divisor
D = E ordy(sJ.V,
where for each irreducible analytic hypersurface V we pick a so that V n U O-
This assignment is a homomoiphism: Let {f}, {1 } denote local defining
functions for D, D respectively. Then
D+Dm (f .f ) € E\Mjt/0*).
This proves a). To prove b) we let D be a divisor on M with local defining
functions f 6 H (U ,jt) relative to an open cover // = {U }. Recall that D is
& & &
identified with (fj 6 H^M^/IT*). Set g^ = ^ • Then
8.b « H^U, n U^, 0*).
We also have g , = gr and in U n U, n U
°ab °ba a b c
ff .£ .a = ^.CU.k = 1
*ab ^bc ^ca fb fc fa "
This means that {g ,} is a cocycle, hence it defines a line bundle L^ €
H {M,0*). To show that L-. is well-defined suppose {('} be another local data
for D. Then
a b a
It follows that the cocycles {g',}, {g ,} are equivalent, hence define the same
line bundle. We also have
since {f •? } are local defining functions for D+D, if {f }, {1 } locally define D,
D respectively. This means the map Div(M) -* H (M,^) given by D h L.. is a
66

homomorphism. We now prove that the kernel of the map D h L_ consists of
principal divisors. Suppose D = (f) for some meromorphic function f on M.
Then we can take as local defining functions for D over any cover // = (U ) the
functions
Then jA = g = 1 and thus L-. is trivial. Conversely, if D is given by {f } and
the line bundle L-. is trivial then there exist functions
h^ 6 HO(Uy)
such that ^f=g^^ = ^- (Use (J) of §2 with g^j^ = 1.) Then
f = f -h"^ = f^.hr^
a a 0 0
is a global meromorphic function on M with (f) = D. a
Two divisors D and D' differing by a principal divisor are said to be
linearly equivalent, and we write D ~ D'. Thus
Div(M)/~ ^ H^(M,(?*).
Let L -♦ M be a line bundle with trivializations (//, y? ). A holomorphic
section s of L -* M over U is identified with a holomorphic function on U via
the trivialization tp . More generally, a holomorphic section s of L -♦ M over
any open set U is given by a collection of functions
{s^ 6 H°(U n U^, 0)}
with s =g,'S, inU nU, nU. A meromorphic section s of L over U is
a ''ab babe ^
simply a collection of meromorphic functions
{s^ 6 H°(U n u^, m
satisfying s^ = g^j^-Sj^.
We leave it to the reader to check that the quotient of two not identically
zero meromorphic sections of a line bundle is a well-defined meromorphic
function on M.
67

If s is a global meromorphic section of L -* M then
Hence for any irreducible hypersurface V c M we have
ord^(sJ = ord^(s^).
We define the order of s along V to be
ord^(s) = ord^(sJ,
where a is any index with U n V i^ 0. The divisor of a global meromorphic
section s of L -* M is defined to be
(s) = E ord (s).V.
V
Clearly (s) is integral if and only if s is holomorphic.
Proposition. Let L -* M be a line bundle. Then L = L_ for some D 6 Div(M)
if and only if L possesses a global meromorphic section which is not the zero
section.
Proof. Suppose L = L_, D 6 Div(M). Let D be given by local data
{f^ 6 H°(U^/)}.
Each f is defined up to H (U ,0*). So we may assume that the collection {f }
& & &
is so chosen that
^ = g , for every a and b,
where {g ,} are the transition functions of L -» M relative to some trivializations
(//, tp ). This means {f } define a global meromorphic section. Conversely, if L
is given by trivializations (2/, (p ) and if s is any nonzero global meromorphic
section of L then
It = gab for every a,b.
So L = L/ V. D
(»)
Note that L € H {Mfl*) is associated to an integral divisor if and only if it
has a nontrivial global holomorphic section.
68

Given a k-dimensional analytic subvariety S c M° we saw in §1 that there
si«K
exists a unique cohomology class [(p] 6 H. (M) satisfying
a = «A y>. for every a € Z. (M).
Js Jm ^ '^
[<p] is the Poincar6 dual of S c M. The Poincar6 dual of a divisor, D = E a.V.,
on M is simply
E a.[v,.] 6 h2(M),
where [tp.] denotes the Poincar6 dual of V..
Proposition. The Poincar6 dual of a divisor D is the Chern class of the
associated holomorphic line bundle L-^.
For a proof of this see [GH] pp. 142-143.
TM (respectively, T*M) will denote the holomorphic tangent bundle
(respectively, holomorphic cotangent bundle) over M. The canonical bundle on
M is the line bundle
K-, = A"T*M.
M
R {M,0{K^,)), the holomorphic sections of Kj^ -* M, consists of holomorphic
n-forms on M. For M = P° we find that
Km = [HI = (H*)*("-^^),
where H -» P" denotes the line bundle associated to a hyperplane divisor and
a; = i!:i A • •. A ^°, y. = z./z., i M-
yi yn ''i y 0'
To see this just note that u) has a single pole along each hyperplane z. = 0.
In general we can compute the canonical bundle K„ of a smooth analytic
hypersurface V c M by the adjunction formula-.
Proof. The normal bundle on V is defined to be the quotient bundle
N^ = TM|^/TV.
We claim that N* • L„|„ is the trivial bundle on V: Let V be given locally by
69

functions f € 0{\J ). Then the line bundle L^ is given by transition functions
& & V
{g , = f /f,}. Since f = 0 on V n U , the differential df is a section of the
conormal bundle N^. (N^ is the dual of N^, i.e., it is the subbundle of T*M|„
consisting of cotangent vectors that are zero on TV c TM|„.) Since V is
smooth, df is everywhere nonzero. On U n U, n V,
Thus the collection (df ) gives a nowhere zero global section of N* • L„|^,
hence N* • Ly|„ is the trivial line bundle on V. This establishes the claim.
Coming back to the main proof we see that
N "^ L I .
Consider the exact sequence of vector bundles over V given by
0 -» N* -» T*M|^ -» T*V -» 0.
By linear algebra we find that
(A°T*M)|y K A°"Vv»N*,
i.e..
An important theorem of Kodaira-Spencer says that for a projective M
Div(M)/~ = H^(M,(?*),
i.e., every line bundle is associated to a divisor on. a projective manifold. For a
proof of this theorem see [GH] p. 162.
, Sometimes we use the same symbol to denote a divisor and the line bundle
associated with it. For example, K will denote the canonical line bundle or a
canonical divisor.
Given a diviaor D on M we put
L(D) = (f € H°(M,/): (f)+D > 0} U {0}.
We also put
|D| = {E € Div(M): E > 0, E ~ D}.
70

|D| consists of all integral divisors linearly eqmvalent to D.
Note that
|D| ^IP(L(D)),
where P(L(D)) denotes the projectivization of the complex vector space L(D).
Proof. Suppose D e |D|. So there exists a meromorphic function f on M such
that D = D+(f). Now f € L(D) since D > 0. Suppose
D = D + (fj) = D + (f^)
for f , f 6 H (M, Jit). Then f. must be a nonzero constant multiple of fj since
the zero sets and polar sets of f. and f coincide: f. = \-i., where A is a
nonvanishing holomorphic function on M, hence A = c ^ 0. a
We record the isomorphism
|D| -» IP(L(D)), D = D + (f) H {Af: A € C}.
Fix a global meromorphic section ^ of L = L_ -♦ M. If r/ is any
holomorphic section of L -♦ D, then r//^ is a meromorphic function on M. (The
quotient of any two nontrivial meromorphic sections of L -* M is a well-defined
meromorphic function on M.) Moreover,
iv/O ~ iv) - (0 > -D Since (0 = D.
Consequently
T,/^ 6 L(D), and (r,) = D + (,,/0 6 |D|,.
Conversely for any f 6 L(D) the section j; = f*^ of L -► M is holomorphic.
Summarizing we have the isomorphisms
xf- L(D) ^ H°(M, (?(Lp)), fHf.^;
(-0"': hO(M, 0{L^)) -. L(D), T, M rj/(.
If L = L-. is a line bundle associated to a divisor D, then we put
|L|=|D|.
Note that |L| is also the projectivization of the space of holomorphic sections of
L -► M, i.e.,
71

|L| = JP{e\m, 0{L)).
A linear ayatem on M is a projective subspace of some |D|. So, a linear
subsystem of |L| = |D| is of the form P(E), where E is a subspace of L(D), or
a subspace of H (M,0(L)). We will write |E| and P(E) interchangeably. A
linear system of the form | D | is called a complete linear system.
The base locus of a linear system is defined by
Base(|E|) = {p € M: p € support(D') for every D' € |E|}.
Let A = |E| be a linear system on M. By way of notation
A = A°, d = deg(A), n = dim(A).
Let A = A° c IDI be a linear system on M. Via the isomorphism
|D| -»IP(L(D)), D + (f)M {Af: X 6 C}
we may identify A with a linear subspace of L(D). We often do this without any
explicit mention. Now fix a meromorphic section (not the zero section, but
otherwise arbitrary) ^ of L_ -* M. We then obtain a linear subspace
K^(A) = {{'(: f 6 A C L(D)} c Am, <7(Ljj)).
Observation. Let Lj. -» M be a line bundle associated to a divisor D, and also
let ^ be a nontrivial meromorphic section of L-. -* M. Then a point p € M is a
base point of A c |(0I i^ ^^^ only if all sections in x^(A) vanish at p.
Proof. All sections in «^(A) vanish at p € M if and only if
(t) f-^(p) = 0 for every f € affine(A) c L((0),
where affine(A) denotes the subspace of L((^)) corresponding to A under the
isomorphism
D = (0 + (f) 6 A M {Af: A € C} € IP(L(0).
The divisor (f • 0 is integral and (f) holds if and only if
(p)<(f.0-(f) + (0 = D6A.
(Keep in mind that (f* 0 is a holomorphic section of L_^ -* M.) But this means
that p is in the support of every divisor 6 A. a
72

A minor technical point to make is that for linearly equivalent divisors D
and D the sets |D | and |D | are equal, whereas L(D ) and L(D ) are only
linearly isomorphic.
A linear system A = | E | c | D | is said to be base-point-free if Base( | E |)
= 0, i.e, the common support of divisors in E is empty (or |E| < IP(H (M,0(L))
has no base points if not all s € E vanish at any p € M).
The fundamental fact concerning linear systems of divisors on a compact
complex manifold is the following: there is a natural correspondence between
base-point-free linear systems on M and nondegenerate holomorphic maps
N
M -► IP , N = the dimension of the linear system,
modulo projective automorphisms.
For any p € M we put
Note that
Hp= |{8eE:s(p) = 0}| < |E|.
H = |E| iffp € Base(|E|).
Assume that |E| is base-point-free. We can then holomorphically embed
M into P^, N = dim|E|, as follows: Pick a basis 8q,---,Sj^ of E c H°(M, 0(L)).
Given trivializations {U, <p ) oi L -* M vre identify
We then map
ig:M^ipN, pGU^H[Sj,^,...,sj6lpN.
The linear system |E| can be recovered from the hyperplane sections of
igCM) C P^,
|E| = {H|./j.y H is a hyperplane divisor on P }.
Observe that i(M) is nondegenerate, i.e., no hyperplane in P contains i(M); a
change of basis for H (M,0(L)) would change i„ by a projective automorphism in
PGL(N+1,C).
73

We have
H = {the hyperplane divisors on i(M) through i(p)}
= {the hyperplanes in IP through i(p) restricted to i(M)}.
N
Now the space of all hyperplanes in IP through a fixed point has dimension N-1,
and it follows that H is a hyperplane in |E|.
We may think of i„ as
E
i„- p H H € lEl* ~ P^.
cj P
N
Let H -* IP denote the hyperplane bundle. Then
i^CH) = L.
Now c^(L) € H^^(IP^), and we can identify hJ^(IP^) with I. Then
deg(ig(M)) = cN(L) = c^(L)[M].
Remark. Let a; be a real type (1,1) form on M. Write
u) = i h..(z) dz' A dz^
where (z') are local holomorphic coordinates on M. The form u is called a
positive form if (h..) is positive definite Hermitian everywhere. A line bundle L
on M is said to be positive if its Chern class can be represented by a positive
form. The famous Kodaira embedding theorem says that given a positive line
bundle L -* M there exists an integer k such that for any k > k. the map
i,,,,:M^lpN
is an embedding.
§4. Hodge Theory for Kahler Manifolds
Let M be a complex manifold. Its complexified cotangent bundle
decomposes as
T*M = T*M • C = T^% 9 T°'^M.
We often write T M instead of T-^M. There is the complexified exterior bundle
74

over M given by
A(T*M) = ®S A*(T*M),
and the decomposition
A*(T*M) = ®E AP'''(T*M).
p+q=a
The space of sections of A*(T*M) -* M is denoted by A*(M), or just
A*(M). Also the space of sections of A'*'*'(T*M) -* M is denoted by A'*'*'(M), or
using the sheaf notation
Recall the (p,q)-th Dolbeault cohomology group of M given by
d d d
The Dolbeault isomorphism states that
' d
where fl^ denotes the sheaf of germs of holomorphic p-forms on M.
Let (M, ds ) be a compact Hermitian (not necessarily Kglhler) manifold.
Also let (^*) be a local unitary coframe on M. The KShler form is given by
We will make AP'*'(T*M) (the space of type (p,q) forms on the complex vector
space T M) into an inner product space as follows: Decree that the basis
{<p A <p : 1,3 multiindices with |I| = p, |J| = q}
is an orthogonal set with | A^''l^ = SP"*"**. (Recall that |dz|^ = 2 on C.) For
<T,T £ AP'*'(M) we put
«T,T> = I <<7(z),r(z)> dvol(z),
where dvol = a/*/n! is the volume form. With this inner product AP'*'(M)
becomes a pre-Hilbert space.
The Hodge star operator
*: AP'''(M) -» A°-P'°"^(M)
75

is determined by the following requirement: if r € A'*'*'(M), then for every a €
A'*'*'(M) we must have
<tr(z),r(z)> dvol(z) = a{%) A *r(z) 6 A°'°(M).
Note that
dvol = *1, **r = {-\)^'^\^ and
*: A°'°(M) = C -» A°'°(M).
Put
a* = -(*oflo*): AP'''(M) -» AP'^^'^M).
It is easily verified that for any a 6 A^'^l'^M), r € AP'*'(M)
<fl<7,r> = <<7,fl*r>,
that is to say, b is the adjoint of fl. Note also that since fl = 0 we also have
8* = 0. (Of course, A'*'*'(M) is not a Hilbert space (it is not complete), and we
do not know that ~b is bounded. So ~b* is a formal adjoint of ~b only.)
Observation. A 5-dosed form a € Z'*'*'(M) is of minimal norm in the set
'b
a + fl(AP'''"^(M))
if and only '\{~b*a ~ 0.
Proof. If 8*<7 = 0, then for any r € AP'*'~^(M)
|(7 + flr|^ = <<7+flr,<7+flr>
= |<t|^ + l^r|^ + 2Re«7,flr>
= |a|^ + l^r|^ + 2Re<fl*<7,r>
= |a|2+ |-flr|2> |a|2.
Hence a is of minimal norm. On the other hand, if a is of minimal norm, then
(fl/5t)(|<7 + tflr|^)(0) = 0, for any r € AP'''"^(M).
Rewriting we obtain
2Re<<7,flr> = 0.
Considering ir instead of r we also obtain
76

21m<a,dT> = 0.
So, <o-,8r> = <8*o-,r> = 0. Since r was arbitrary this finishes the proof, a
Remark. Let a € AP'*'(M) and put
S = (7 + flAP'**~HM).
Consider the Hilbert space completion Hil'*'*' of A'*'*'(M). We then know from
Hilbert space theory that there exists r of smallest norm in the closure
cl(S) c HilP'^
The Hodge theorem says that this element is in fact in A'*'*'(M).
The above argument suggests that the (p,q)-th Dolbeault cohomology
group is represented precisely by the solution set of the equations
da = d*a = 0.
These two equations can be combined into a single second order equation
where
A_(7 = 0,
d
A_ = ~do8* + 8*o~d: AP'^'CM) -» AP'^'CM)
d
denotes the fl-Lapladan.
The space of harmonic type (p,q) forms on M is
lP'<i(M) = {a € AP'^^CM): A a = 0}.
d
The Hodge theorem states that
dim TP'^'CM) < B,
and, moreover,
d
For a proof of the Hodge theorem we refer the reader to [GH] pp. 84-100.
The Hodge isomorphism combined with the Dolbeaul* isomorphism give
r^'^^iU) ^ H''(M,nP).
In particular.
77

afP'°(M) ^ H°(M,nP).
That is to say, holomorphic p-forms are fl-harmonic.
Since *oA_ = Ao^c the Hodge star operator induces isomorphisms
d d
We thus obtain (metric-dependent) isomorphisms
H''(M,nP) ^ H°-^(M,n°~P).
Eemark. There are purely topological isomorphisms
H''(M,nP) % (H°-^M,n°-P))*
coming from the exterior (or the cup) product.
The numbers
hP'** = dim T[^''^{M) = dim H''(M,nP)
are called the Hodge numbers of the compact Hermitian manifold M. We have,
so far,
I) hP'** < a.; h°'° = 1; hP'** = h°-P'°-^.
The Kunneth formula gives
h*'^(M K N) = E hP'''(M).h'''(N),
where the sum is taken over p,q,r,s with p+r = a, q+s = b.
We now suppose that M is a compact Kahler manifold. (We have in mind
smooth projective varieties.) An important technical consequence of the K&hler
condition is that
(t) Aj = = 2/i, = 2A
0
The reader should try a proof of (f).
As a consequence of (f) we see that A. preserves the type of a form.
Some of the well-known topological properties of a compact Kahler
manifold M° are;
a) h^ > 0, h^i'*! > 0;
78

b) the underlying homology class of an analytic subvariety is nonzero.
c) a nonzero holomorphic form is closed but never exact.
Proof. We give u^ {u = the Kahler form) as a closed 2q-form, in fact type
(q,q) form, that is not exact. This will prove a). If we had u = dij, then
f (/ = f d(T7 A (f-^) = 0.
But this is absurd as (J^-nl = dvol. To prove b) let V** c M° be a
d-dimensional analytic subvariety. Wirtinger's theorem says that
Vol(V) = (l/d!)f i/ i 0.
So the linear functional
1
: Hf (M) -* R
V °
is nonzero. Poincar6 duality does the rest. We leave the proof of c) to the
reader as an exercise, o
Put
RP'^^CM) = RP'^^CM) = ZP'''(M)/{dA*(M) n ZP'^'CM)}
(HP'*'(M) consists of d-closed type (p,q) forms modulo d-€xact type (p,q) forms
on M),
TP'*»(M) = {a 6 AP'*»(M): A^a = 0},
Tj(M) = {a 6 A'(M): A^a = 0}.
Since A . = 2A we obtain
'^ ~d
** d '^ 8
Proposition. Let M be a compact Kahler manifold. Then
p+q=a
2b) Ttl'^^iU) = f3'P(M).
3) HP'^^CM) = 'Xl''^{M).
Proof. 1) follows from the from the fact that A. preserves the type, i.e..
79

where tt^'**: A_(M) -* A'*'*'(M) is the type projection. 2) follows from the fact
that Aj is real. To prove 3) take r/ € Zj'**. Then
V - dd*G(T,) 6 tP'^
where G denotes Green's function, a
The Hodge theorem says that
(4) H;(M) = r^{M).
For a compact Kahler manifold we have the Hodge decompositiom
a) H'(M,C) = ®E HP'^^CM);
p+q=a
b) HP'^^CM) = H'i'PCM).
Proof, a) follows from 2a), 3), and 4). b) follows from 2b), and 3). a
In terms of the Hodge numbers of M we have, in addition to (I),
(II) hP'** = h'i'P; b^ = E hP'**; hP'P > 1.
p+q=a
Note that
HP'°(M) K TP'°(M) % H°(M,nP).
Thus on a compact complex manifold a holomorphic form is harmonic with
respect to any Kglhler metric.
An immediate consequence of the Hodge theorem is that the odd Betti
numbers of a compact Kahler manifold M are even
To see this just note that
Vi(M) e 21:
b, ^, = 2 E hP'^'i+l-^.
2*1+1 p=0
CoroUary. H'i(IP°,nP) = C if p = q; 0 if p M-
Proof. Since H^^+V") = 0 we have
HP.<i(|pn) = 0 for p+q odd.
80

Also H^\lP°,ff) ^ I. So for p ^ k,
1 = b^^ > hP'2^-P + h^^-P'P = 2hP'21^-^.
Hence hP'^^'P = 0. So
HP'P(IP°) ^ hJp(IP°,C) ^ C. d
8
Consider a holomorphic rank r vector bundle E over a compact complex
manifold M. An E-valued smooth complex type (p,q) form on M can be given
locally as
a = a'* 9 e ,
a'
where (e ) is a local holomorphic frame for E, and a** 6 AP'*'(U), U an open set
in M. We let AP'*'(E) denote the space of all such forms. Although the exterior
operator d: A'(M) -* A'"*" (M) does not naturally extend to A'(E) we do have
~8 = \: AP'^^CE) -» AP'^i+^CE)
given simply by
8{a) = Sa** 9 e^.
To see that this operator is well-defined let e = (e ) be another holomorphic
frame. Then e and e are related by
for some GL(r,C)-valued local holomorphic function g. So
<T = g^o-** • §-, and
8<T = 8{g<T) • e = g8<T 9 e = 8a 9 e
since dg = 0. Note also that d^ = 0.
We let ZP'*'(E) denote the space of 5-dosed E-valued smooth
d
complex-valued type (p,q) forms on M. Since fl = 0 we obtain cohomology
groups
RP'^^E) = ZP''i(E)/flAP'''"^(E).
d 8
The Dolbeault isomorphism for holomorphic vector bundles is
81

HP.q(E) ~ H'iCM.nPCE)),
d
where nP(E) denotes the sheaf of germs of E-valued holomorphic p-forms on M.
To do Hodge theory on E we choose Hermitian metrics on E and M. Let
{<p^) be a local unitary coframe on M; (e ) be a local unitary frame for E. Any
section a € A'*'*'(E) can be written locally as
a(z) = (l/p!q!) a^/z) v>^ A i^^ A e^,
where |I| = p, |J| = q. Define an inner product in A^'*'^(E) by
«T,T> = I «t{z),t(z)> dvol(z),
where
<a(z),r(z)> = (2P+''--/p!q!) a^ .,(z).r^ j(z).
A'*'*'(E) is now a pre-Hilbert space. We also have
*g: AP'^^CE) -» A°-P'°-^(E*)
given by
,^ia) = (*0 . e^,
where (e*) is the dual unitary frame on E*. We now define
~d* = ~dl = -{*e°^°*e)= ^'''"^^'^ ^ AP'^^-^E).
We then verify that
<\<T,r> = <a,fl*r>, a € A^''^-\ r € A^'\
i.e., 5* is the adjoint of fl„.
The ^--Laplacian is given by
The Hodge Theorem for Hermitian Holomorphic Vector Bundles. Let E -» M be
a Hermitian holomorphic vector bundle over a compact Hermitian complex
manifold. Then
1) dim Tt^'^^iE) < m;
2) TP'^(E) ^ HP'^^CE);
d
82

3) *„ induces isomorphisms
H'i(M,nP(E)) ^ (H°-^(M,n°-P(E*)))*.
Since n°(E*) = 0{E* • K ), the sheaf of germs of E*-valued holomorphic
n-forms on M, we have
E\M,0{E)) ^ (H°-^(M,(?(E* • Kj^)))*.
This isomorphism is called Kodaira-Serre duality. Taking E to be K we obtain
h'(M,(?(K^)) = h"-^(M,(?^).
Recall that a (holomorphic, as always) line bundle L -* M (M, any
compact complex manifold) is positive iff its Chern class € Hj(M,IR) can be
represented by a real positive (1,1) form. That is to say, the Chern class of a
positive line bundle has a representative 2-form which is the Kahler form of some
Hermitian metric on M. A line bundle L -* M is negative if L* is positive.
Proposition. The hyperplane bundle H -* IP°, is positive.
Proof. The Chern class C-(H) is represented by the Kahler form of the (suitably
normalized) Fubini-Study metric on IP°. (See [GH] p. 150 for more.) a
The line bundles [mH] -♦ IP° are positive for m > 0 and negative for m <
0. Given a smooth variety i: M c IP° the puUback bundle i~ H -* M is also
positive.
Observation. Suppose L -* M° is a positive line bundle. Then Cj(L) > 0.
Proof. Represent c.(L) be a real positive (1,1) form w on M. Then
c°(L) = c°(L)[M] = f (/ = n!.vol(M) > 0,
where vol(M) is computed with respect to w. a
For a proof of the following theorem we refer the reader to [GH] pp.
154-155.
The Kodaira Vanishing Theorem. For a positive line bundle L -► M,
H''(M,nP(L)) = 0, if p+q > n.
83

That is to say, there exist no nonzero harmonic L-valued forms of degree larger
than n.
Dualizing we obtain
(a) H''(M,nP(L)) = 0 for p+q < n, if L -» M is negative.
Corollary. H*'(IP°,0(kH)) = 0 for every k, where 1 < q < n-1.
Proof. If k < 0, then (a) gives the proof. Suppose k > 0. Then
H''(IP°,C'(kH)) = H'i(IP°,n°(kH-K)) = H'i(IP°,n°((k+n+l)H)) = 0. a
§5. Bundles over Complex Projective Space
From the cell decomposition
pn ^ (|pn\|pn-l) U (|P°-l\|P°-2) fl ... U {a point}
we find that
H2.(IP°,2) ^2 for 1 < i < n;
H^-^lCPMr) = 0.
Recall from the preceding section that
H*'(IP°,nP) =: C, if 0 < p = q < n; = 0 otherwise.
In particular,
H°(IP°,nP) = 0,
i.e., there are no global holomorphic p-forms on IP°.
For every i, H'(IP°,0) - 0. To see this just note that
tf(IP°,(?) = H'(IP°,(?(1)),
where 1 = Ip^ denotes the trivial line bundle, and 0{1) denotes the sheaf of
germs of its holomorphic sections. In other words,
tf(IP°,(?) = tf(IP°,n°).
But H'(IP°,n°) ^ H°(IP°,n') = 0.
As a consequence
84

H^(IP°,0*) K H^(IP°,ff) % 1,
i.e., a holomorphic line bundle on IP° is determined by its Chern class.
Take a hyperplane H. c IP°. It defines a homology class in H (IP°,2)
which we again denote by H.. The Poincar6 dual of H- will be denoted by r/ €
H (P ,ff). The Poincai6 dual of a hyperplane is independent of the particular
hyperplane chosen since all hyperplanes are homologous to each other, and r; is a
generator for H^(P°,ff).
The line bundle associated to a hyperplane, the hyperplane bundle, is
denoted by H. All hyperplanes are linearly equivalent to each other as divisors
so that H is well-defined. We have
Cj(H) = «(H) = ,; 6 hV.2)-
Since T] generates H (P ,1) we see that any line bundle L is of the form
L = H*™ = mH for some m € ff.
For example,
K = -(n+l)H,
where K -* P° is the canonical bundle.
The universal bundle, S -► P°, is the subbundle of P° x C""*"^ given by the
following prescription: the fibre at [x] = I 6 P° is the line I C C°"*" .
Proposition. S = H* = -H.
Proof. Define
\-' Uo = {xj, * 0} C P" ^ S
by
where x = (x ,'«',x ) are the homogeneous coordinates. The map s. is
holomorphic and nonzero in U.. Moreover, it extends to a meromorphic section
over P°. Indeed,
85

It follows that S is the line bundle associated to the divisor -H.. o
The fibre at [x] = /^ € IP° of H is
(/ )*, the space of linear functionals on / .
We claim that H°(IP°,(?(H)) is naturally identified with C°+^* = Hom(C°+\c).
Any f 6 C , simply by restriction, induces a section
a^6HV",(?(H)):a/x) = f|,^.
Clearly a^ = 0 only when f is. Thus we have C""*"^* included in H°(IP°,(?(H)). To
see that the assignment is surjective take any a € H , and put D = (a). Now
c-(H) is the cohomology class of D, and since any variety homologous to a
hyperplane is a hyperplane we conclude that D is a hyperplane divisor. Let f 6
C""*" * be any linear functional vanishing on the hyperplane t~ D c C""*" . Then
the meromorphic function a/a. will be holomorphic on all of IP°, hence constant.
Consequently, a =: cc- We have just proved
H°(IP°AH)) = C°+^*.
More generally, we have identifications
H°(ip°,c;(H'*)) = s'^cc"-^^*),
where S'^CC"'^^*) denotes the space of homogeneous polynomials of degree d in
On 0
n+1 variables. For example, JP(H (IP ,0{R))) can be thought of as the linear
system of quadric hypersurfaces in IP°.
Example. The Veronese map is the embedding
i: JP° -f IP^, N = dim IP(H°(IP°,C;(dH))) = C*+°) - 1,
associated to the complete linear system |dH|. For example, the Veronese
surface is given by
i^^j^y^^^^\ [l,8,t] H [I,8,t,s2,8t,t2]
in terms of the inhomogeneous coordinates s = x /x , t = x-/x . The degree of
the Veronese surface is Cj(2H) = 4.
We know that if V C P" is an analytic hypersurface, then for some d € ff
86

since V is a divisor. It follows that V is the zero locus of a section in
H°(IP°AdH)), i.e.,
where f =: f(x , • • • ,x ) is a homogeneous polynomial of degree d. More generally
there is the
Chow Theorem. Any analytic variety of IP° is an algebraic variety.
Chow's theorem is a manifestation of a rather general principle in complex
algebraic geometry known as the G.A.G.A. principle (named after Serre's paper
"Geometrie algebrique et geometrie analytique") which asserts that a global
analytic object on a projective variety is algebraic.
The Holomorphic Tangent Bundle over IP"
Let TIP" = T^'V denote the holomorphic tangent bundle over IP°.
Consider the holomorphic projection
T. c°+^{o} -»ip°.
Put y. = x./x , i ^ 0. (y.) are the inhomogeneous coordinates on IP'^\IP°~ , where
°~ is the hyperplane at infinity defined by x
IP° = IP° ^ is the hyperplane at infinity defined by x = 0. We compute that
T*dy. = (xpdx. - x.dxp)/xj
So at a point x 6 C""*" we have
Tjfl/to.) = (l/xp)(5/fiy.), i f 0;
T^(fl/5Xj,)=-S(x./x2)(5/5y.).
Note that
(t) %(S X.(fl/ax.)) + T^(Xp(fl/5Xp)) = 0,
where we think of x„,x.-. C""*" -* C as linear functionals.
0' 1
Remark. Take a linear functional a: C""*" -♦ C and consider the type (1,0) vector
field
87

v(x) = o{x){d/dx.).
Now T^(v(x)) = T^(v(Ax)), A € C, X € C"''"^ Hence
%(v(x)) = vl[x]
makes sense.
Tr ilP° is spanned by
{T^(fl/ax.), 0 < i < n}
with the single relation (f).
There is a bundle map
$: H®(°+^) = H • ... • H -» TIP",
where a. 6 H (IP°,0(H)). The map * is certainly surjective, and Ker(*) is the
trivial line bundle spanned by the section (x ,'",x ), We thus obtain so called
the Euler sequence
0 -» ip^ -» H®(°+^) -» TIP" -» 0.
And from a smooth decomposition (introduce a metric)
H®(n+l) = TIP" 9 lp„
we obtain
c(H®(°+l)) = c(TIP°).l
= c(H)...c(H) = c(H)°+^
= (1 + Cj(H))"+l = (1 + „r+\
where c denotes the total Chern class. It follows that
c(TIP°) = (1 + 7/)°+^ € H^*(IP°,ff).
Another way to think of a tangent vector to IP° at a line / c C""*" is to
view it as a linear map from / to C""*" jl — J^ (choose an inner product). The
universal quotient bundle Q -* IP° is defined by the exact sequence
0 ^ S - (n+l)p„ - Q - 0,
where (n+l)p„ = p" x C"''" . Q ^ S"^ with respect to a Hermitian metric in
88

(O+l ^g thiig obtain identifications
T^'V ^ Hom(S,Q) ^ HomCS.S-") ^ S* • S\
Tensoring the above exact sequence with H = S* we obtain
0 ^ lp„ - H • (n+l)p„ -. TIP" - 0.
Again the Whitney product formula gives
c(TP") = c(H • (n+l)pj.l = c(H®("+^)) = (1 + 7,)"+^
Proposition. Cj(S"^) = Cj(H) = r/.
Proof. Since TIP° = H • S""
Cj(TIP°) = Cj(S^) + rank(S^)Cj(H). a
Hermitian Geometry: Unitary Frames
Let e = (^Q.-'-.e^) ^ U(n+1). U(n+1) acts on IP° by
e-[x] = [e-x].
Consider the ilbration
r. U(n+1) -» IP°, e H [e^].
The projection t is a principal U(l)«U(n)-bundle, and the isotropy group of the
above action at the origin = [f ] = [ (1,0,-".O)] € IP° is
0'
U(l)xU(n) = Gp = {Ql): a 6 U(l), A 6 U(n)}.
Put
m =
0 *X
, X€ C°
H C°, (•] H X.
X 0
The vector space m is the orthogonal complement (with respect to the Killing
form) to i)y the Lie algebra of G.:
g = f) © m.
Let fl denote the Maurer-Cartan form of U(n+l). It is the u(n+l)-valued
left-invariant 1-form on U(n+1) given by
flg.- TgU(n+l) - T.jU(n+l) = g, v m L^.^^v.
89

The Maurer-Cartan form fl decomposes as
I) m
With respect to the natural basis (f ) of m = C° we write
n„ = n^ • c, 1 < a < n.
m 0 a' - -
Note that s*{CIq) span T*^'V, where s is a local section of U(n+1) - IP°.
By way of notation we put
s*n = u).
The Fubini-Study metric on IP° (with holomorphic sectional curvature 4) is
given by
The Kahler form is
ds^ =: E w^ • a;^
0 0
5 S o^J A 0,°.
Using the Maurer-Cartan structure equations
dn = -n A n
we compute that
i.e., {(Jt - S^oJq) = ^ are the connection forms. Now the curvature forms are,
by definition, given by
X = d^ + ^ A ^.
So
and from this we see that the holomorphic sectional curvature is 4.
The Chern classes of TIP" can be computed explicitly in terms of the
curvature forms x- For example.
We know that
90

C(IP°) = (1 + T/)°+^
So
(1 + 7,)"+^ = E C..
1=0
In particular,
(n+l)T7 = Cj.
Summarizing
(n+l)Cj(H) = Cj(TIP") = [^i trace(x)].
We will compute the Chern classes of Grassmannians and projective
hypersurfaces in Chapter V.
91

Chapter III. Curves and Compact Riemaim Surfaces
Standard materials on the theory of compact Riemann surfaces and
projective realizations of them are covered in this chapter.
§1, which can be read separately from the rest of the chapter, deals with
singular plane curves; the classical Pliicker formulae are given. In §2 we discuss
preliminary results from function theory: the equidistribution property, the
Riemann-Hurwitz formula, the residue theorem, etc. Linear systems of divisors
on a compact Riemann surface and the corresponding holomorphic curves in
projective spaces are discussed in §3. Examples including the rational normal
curve and the Weierstrass nonsingular plane cubic are also given in the section.
In §4 a discussion of Torelli's theorem yielding a parametrization of Riemann
surfaces of genus g by the Siegel half space modulo Sp(g,ff) is given. Abel's
theorem and the Jacobi inversion theorem are also proved. As an application we
will see that the space of all nondegenerate holomorphic maps from IP to IP" of
degree d is, more or less, the complex Grassmannian G(d+l,n+l). A theory of
hyperelliptic surfaces via the Weierstrass points is given in §5. A proof of the
Riemann-Roch theorem and the standard results on projective embeddings are
given in the final section.
For materials directly pertaining to Riemann surfaces we recommend [FK]
as a supplementary reference to this chapter. The book [N] gives a good
treatment of curves balancing different perspectives.
Important but advanced topics such as theories of moduli and the
Brill-Noether theory are not covered. The recent book [ACGH] gives an
authoritative account of the Brill-Noether theory. For the problem of moduli the
reader may consult the article [Ha] and many references cited therein. For a
survey on theta functions and related topics the reader may consult [Gu2].
92

§1. Plane Curvefl
In this section we consider algebraic curves in IP . Theory simplifies a
great deal due to the fact that a plane curve is also a hypersurface.
Topologically speaking an algebraic curve is a connected compact oriented
real two-manifold with finitely many points identified; these are the topological
singular points. (A topological singular point is always singular, i.e., nonsmooth.
although a singular point may not be topologically singular.) For a smooth curve
in IP the genus is determined by the degree, and vice versa. This is in marked
contrast to the case of space curves: a nonsingular quartic curve in IP can be
either an elliptic curve (genus 1) or a rational curve (genus 0).
Let f e C[x,y,z] be a homogeneous polynomial. Then
V(f) = {[x,y,z] e P^: f(x,y,z) = 0}
is a curve and every curve in IP arises in this way. Since C[x,y,z] is a unique
factorization domain f factors into
i = f^i...r^(m. > 1),
1 n ^ 1 - '*
where each f. is homogeneous and irreducible. V(f^*) is the i-th irreducible
component with multiplicity m.. In particular, an irreducible curve in IP is
defined by an irreducible homogeneous polynomial in C[x,y,z].
Degree two plane curves are called conies and there is essentially one conic.
Proposition. Let C., C be irreducible curves of degree two in IP . Suppose we
are given distinct points Pj,p-,p, € C. and distinct points q^.q^.q, € C . Then
there exists a projective transformation A e Aut(IP^) = PGL(3,C) such that
A(p.) = q., 1 < i < 3,
and A maps C. isomorphically onto C .
The above proposition is an easy consequence of the following well-known
result from Inear algebra: The only GL(n,C)-invariant of an nxn complex
symmetric matrix is its rank, i.e., any nxn complex symmetric matrix is similar
93

"^ [o'o]'
to the matrix L"^ A, where r is the rank. Observe that an irreducible conic in
2
IP is given by the homogeneous polynomial
Q(x,y,z) = E Q.yxJ (x^ = x, x^ = y, x' = z),
where (Q..) is a complex symmetric matrix of rank 3. In particular, an
2
irreducible conic c IP is smooth.
Let W. denote the complex vector space of all homogeneous polynomials of
degree d, d > 1, in C[x,y,z]. A basis for W. is given by
B = {xVz^ i,j,k > 0, i+j+k = d}.
It follows that
dim(Wj) = id(d+3) + 1.
Let f,g e Wj. Then V(f) = V(g) if and only if f = Ag for some A e C*. It
follows that the totality of projective plane curves of degree d is naturally
identified with IP(W.), the projectivized W..
Example. Have P(W^) ^ IP . On the other hand, as we shall see later, a
smooth quartic curve in IP is of genus 3 (by the genus formula) and is its own
canonical curve; any nonhyperelliptic Riemann surface of genus 3 is canonically
embedded in IP as a smooth quartic. From this we surmise that the moduli of
compact Riemann surfaces of genus 3 is 14-dimensional.
A projective n-subspace of IP(W.) is called a linear system of plane curves
of degree d and dimension n.
Proposition. Let Pj,"',p € IP be distinct points with m < id(d+3). Then
S = {f 6 IP(Wj): V(f) passes through every p.}
is a linear system of dimension ^d(d+3) - m.
Proof. For any p e P^
S = {f e IP(Wj): V(f) passes through p}
is a hyperplane in IP(W.). Now
d^
s = s n ... n s .
PI Pm
94

Hence S is the intersection of m hyperplanes and is itself a projective subspace of
dimension at least id(d+3) - m. We leave the rest to the reader, a
In fact the above proof shows that given any distinct id(d+3) points in IP
there is a curve of degree passing through all of them. Generically, such a curve
is uniquely determined as we shall see below.
Listing the monomials in the basis B in a fixed order we get the Veronese
map given by
*: [x,y,z] e P^ H [ ..., xVz'', • • • ] e P^, N = id(d+3).
We leave it to the reader to verify that the Veronese map is a
nondegenerate holomorphic embedding.
Proposition. For distinct points Pi)"')P« € P , N = 4d(d+3), there is a unique
curve of degree 2 passing through every p. if and only if the points
*(Pi))* • •)*(Pm) £ P *re in general position.
N 2
Proof. A hyperplane H c P determines a curve C c P of degree d, and vice
versa:
H e P^* M C = *"^(H) = *"\h n $(P^)).
The result now follows since the points *(p,),"-.^Cp ) determine a hyperplane
N
in P if and only if they are in general position. D
We now give an explicit geometric interpretation of the degree of a plane
curve. Take an irreducible curve C C P given by an irreducible homogeneous
polynomial f(x,y,z) of degree d. We can write
f(x,y,z) = f/ + fi(x,y)z'*-^ 4- • • • + fj(x,y) (f^ i 0),
where each f.(x,y) is either 0 or homogeneous of degree i. Fix a point p = [a,b,0]
e P . Let L denote the projective line through p and [0,0,1]; let L denote
the projective line through [1,0,0] and [0,1,0]. Consider the equation
(t) i/ + fi(a,b)z'*-l + ... + fj(a,b) = 0.
The above equation has d solutions counting multiplicity. In fact, for a generic
95

choice of p (to be more precise, p needs to lie on L minus the discriminant
variety) this equation has d distinct solutions. This means that the line L and
the curve C = V(f) intersect at d points, namely at
{[a,b,z]: z is a root of (f)}.
2
Let C c IP be a degree d curve, and p e C be arbitrary. Applying a
projective transformation if necessary we assume that p = [0,0,1] and that C does
not contain the line {z = 0} at infinity as an irreducible component.
Dehomogenize: F(x,y) = f(x,y,l). We can write
(*) F(x,y) = Fjxj) + F^_^j(x,y) + • • • + Fj(x,y) (F^ f 0),
where each F. is either 0 or homogeneous of degree i. The affine variety V(F) c
C is the affine part of C relative to the line at infinity {z = 0}. Upon the
inclusion C C IP the topological closure of V(F) is C. Applying the Jacobian
test we see that p = [0,0,1] is nonsingular if and only if m = m = 1. The
positive integer m is the multiplicity of p e C.
Remark. Let V be any irreducible affine variety in C°. Recall that the
coordinate ring of V, denoted by R_., is the integral domain given by
R^ = t[x\...,x°]/p,
where p is the prime ideal defining V. The rational function field of V is just
the quotient field K„ of R„. The local ring of V at p e V, denoted by 0 , is the
subring of K„ consisting of rational functions defined at p. The maximal ideal
M ot 0 consists of rational functions vanishing at p. One can then show that
the multiplicity at p e V is equal to the complex vector space dimension of M*
modulo M*"*" for sufficiently large i.
Resuming our main discussion we suppose that p = [0,0,1] £ C is a singular
point. This means that the integer m = m given by (*) is greater than 1. The
singular point p is called an m-ple point. Decompose F into linear
96

homogeneous factors and obtain
F (x,y) = (a,x + b^y^* • • • (a x + b y)'"°, E m. = m.
Definition. Maintaining the preceding notation we have:
a) the closure of {a.x + b.y = 0} c P is called a tangent line to C at p,
and m. its multiplicity;
b) if m. = 1 for every i, then p is called an ordinary m-ple point (an
ordinary double point is also called a node);
c) if F is irreducible, then s =1 and p is called a cusp. So at a cusp there
is a well-defined tangent line of multiplicity > 1.
Example. Let f(x,y,z) = -x + y z. Dehomogenize:
F(x,y) = f(x,y,l) = -x' + y^.
Setting F = F = 0 we obtain (0,0). Now
* y
F = F^ + F3 with F^ = y^.
So (0,0) = [0,0,1] is a cusp. Dehomogenizing f relative to y we obtain
f(l,y,z) = z - x'.
Hence [0,1,0] is a smooth point. It follows that [0,0,1] is the only singular point
of V(f).
The reader should verify that a nonsingular plane curve is irreducible.
Digression. Let D be a unique factorization domain (we have in mind D = C[x])
and also let F(t), G(t) e D[t] be polynomials given by
m . n .
F(t) = E a.t\ G(t) = E b.tJ,
i=0 j=0 ■'
where we assume that at least one of a , b is nonzero. Then the resultant of F
m' n
and G is defined to be the following determinant:
97

R(F,G) = det
a , a ,, • •', a.
m' m-1' * 0
m' m—1 * * 0
a I a 4 1
m' m-1*
•••' *o
^n'^n-1' "''\
■ n rows
m rows
n
Let Cj = V(f) and C = V(g) be irreducible curves in IP , and p any point
in the intersection. Applying a projective transformation if necessary we assume
that
p = [0,0,1] e Cj n C^.
Dehomogenize:
F(x,y) = f(x,y,l), G(x,y) = g(x,y,l).
We assume that the line at infinity {z = 0} is neither C. nor C.. Now the
point p corresponds to the origin 0 € C and
ordjjR(F,G)(x) = ordQll(F,G)(y),
where R(F,G)(x) e C[x] is the resultant of F,G e D[y] with D = C[x], and
R(F,G)(y) is given similarly. The above order is nothing but the intersection
number of the pair 0.^0- at p which we denote by # (Cj,C ).
The intersection number # (^^^o^ ^*" *^^° ^® defined as the complex
n 0
vector space dimension of OJC ) (the local ring c C(x,y) at 0 e C ) modulo the
ideal generated by F and G.
For two curves that are not necessarily irreducible the intersection number
is defined simply by linearly extending the definition in the irreducible caSe:
#p(°l"C2,C,) = #^(Cj.C,) + #_(C,,C,).
The intersection number # (C,,CJ is defined to be od if C, and C„ hive
98

an irreducible component containing p in common. We often exclude this case
from the consideration by requiring that the curves intersect properly.
Let p 6 Cj n C,) and suppose that p is a nonsingular point of both C. and
C-. Further suppose that the tangent lines at p to C. and C- are distinct. We
then have the intersection number at p equal to 1. More generally we have
#p(C,,C,) > m^(C,).m^(C,).
Moreover, the equality holds if and only if the curves have no tangent lines
common at p.
Example, a) Let F(x,y) = -x' + y^, G(x,y) = x. Then
R(F,G)(x) == det
R(F,G)(y) = det
1, 0,- X3-
0, X, 0
0, 0, X .
-1, 0, 0,
1, 0, 0,
0, 1, 0,
L 0, 0, 1,
s:
0
0
0-
X , and
= y
So the intersection number is 2.
b) Let F(x,y) = y"* - x and G(x,y) = y. We then have
#o(F.G) = 3.
On the other hand
m^CF) = 2, m^CG) = 1.
Bezout's theorem becomes
Theorem. Let C„ C be curves in P with no common irreducible components.
Then
S #p(Ci.C2) = de5(Cj).defi;(C2).
p
Definition, a) A nonsingular point p of an irreducible plane curve C of degree d
> 3 is called a flex if
99

#p(C,TpC) > 3,
where T C denotes the tangent line. The point p is called an ordinary flex if
#p(C,TpC) = 3.
b) A singular point p e C of order 2 is called a (simple) cusp of order 2 if
#p(C.TpC) = 3.
Example, a) Let F(x,y) = x - y. Then (0,0) is an ordinary flex.
b) Let F(x,y) «= x - y . Then (0,0) is a simple cusp of order 2.
Proposition. On a nonsingular curve C C IP of degree d > 3, there exist at least
one and at most 3d(d-2) flexes.
Proof. Write C = V(f), f e C[x,y,z] an irredicible homogeneous polynomial.
Recall that the Hessian of f is defined to be the determinant of the 3x3 matrix
(fl2f/5xW),
where x = x, x = y, x = z. It defines a curve in IP of degree 3(d-2), called
the Hessian curve. It is routinely checked that a point p e C is a flex of C if it
also lies on the Hessian curve. Bezout's theorem does the rest, d
Let N be a complex manifold and C c N be a compact irreducible analytic
curve. Then by Hironaka's theorem, there exists a nonsingular model
V?: M -» C C N
of C. That is to say, M is a compact Riemann surface and ^ is a holomorphic
map such that tp{M) = C, and
<p: M\<p~\c . ) -» C\C .
^ IT- V Bing/ « sing
is a biholomorphism. Moreover, if y>: M -* C is another nonsingular model, then
M is biholomorphic to M.
In particular, there exists a nonsingular model of any curve in IP°.
Conversely, for any compact Riemann surface and a holomorphic map
f: M -» IP°,
100

its image (f^M) is an algebraic curve by Chow's theorem. We thus have another
way of viewing projective algebraic curves, namely as holomorphic maps from
compact Riemann surfaces into projective spaces.
Consider a nondegenerate holomorphic map
f: M -» IP°,
where M is a compact Riemann surface. So f(M) is an algebraic curve, and all
algebraic curves in P" arise in this way. To avoid redundant considerations and
to simplify the subsequent exposition we assume that the map f is nondegenerate,
i.e., the image f(M) does not lie in any lower dimensional projective space. Fix a
point q £ M. In a neighborhood of p the map f can be holomorphically lifted to
C""*" *, i.e., there is a holomorphic map
f = (f°,...,f°): z£U-C°+^*
such that TTof = f, where r. C""*" * -* IP° is the projection and z is a holomorphic
coordinate centered at q. The map f is a homogeneous representative of f.
Applying a linear transformation £ GL(n+l,C) we assume that
f(o) = (i,o,...,o),
i.e., f(q) = [1,0,-••,0]. Since f*(0) = 0 for i > 0 we can write
(f'(z)) = z'>+Hh'(z)), 1 < i < n,
where a. is a nonnegative integer and (h'(0)) ^ 0. Making another linear change
of coordinates we may assume that
(h'(0)) = (l,0,...,0)£C".
We can now write
(hJ(z)) = z'^+^(6^(z)), 2 < j < n,
with a > 0 and (g^(0)) -^ 0. Recursively proceeding we obtain the following local
normal form for f:
f(z) = [1, z'>+^ + ..., z2+»>+»' +..•,..., z°+'>+' • •+'" +...].
Note that a. is nothing but the branch number of f at q. That is to say,
101

aj + 1 = mp,
the order at p = f(q) e C = f(M) c IP°. We will see later that the number a. is
the branch number at q of the {i-l)-th associated curve of f.
Let C C P be an irreducible curve of degree > 2 (so that it is
nondegenerate) and fix a nonsingular model
yv. M -* C.
The local normal form at p = ip{q) e C is
¥<z) = [1, z'>+l + ...,z2+»>+»'+ ... ].
So the point p £ C is nonsingular if and only if a. = 0.
The Gauss map of <p is defined to be the holomorphic map
/-. M-lp2*, zH[(i,)A(i>')],
where [{<p)^{<p')] denotes the 2-plane spanned by {<p),{<p') € C . To put it
another way, we have
/: z e M H T C.
^ p
The curve C* = ip*{M) is also called the dual curve of C.
Remark. The Gauss map ip* is a pirori undefined at a singular point of ip.
However, at a singular point there are finitely many tangent lines and we take
the tangent line that makes the map <p continuous. (The map ip is defined at
first as a rational map. But any rational map on a Riemann surface is uniquely
extended to a holomorphic map.)
We leave the proof of the following proposition as an amusing exercise to
the reader.
Proposition. Suppose we are given an irreducible nondegenerate plane curve
<p:M^ P^.
Define nonnegative integers a. at a point q e M using the local normal form as in
the preceding discussion. Then
a) q £ M is a nonsingular point of both (p and ip* if and only if
102

^1 = ^2 = 0;
b) q £ M is an ordinary flex (of (p) if and only if
c) q e M is a (simple) cusp of order 2 if and only if
aj = 1, a^ = 0.
It is clear that an ordinary flex of ^ is a cusp of order 2 of ^*, and vice
versa.
Given an irreducible curve fiM) = C c IP it is not hard to see that its
dual C C IP is also an irreducible curve. It is also straightforward to verify
that
a) the dual curve of C* is C;
b) ip*: M -* C* is the nonsingular model of C*.
Theorem. Let yv. M -^ C c IP be an irreducible curve of degree > 2. Then
deg(C*) = 2(g-l+d) - E (m -fl ),
p
where d = deg(C), g = genus(M), m = ord C, s denotes the number of
irreducible branches at p of C, and the sum is taken over all singular points of
C.
Proof. Put d* = deg(C*). Take a general point q e IP^\C with the following
properties: q is not on any tangent lines at singular points of C; d* is the
number of tangent lines of C passing through q. Define a holomorphic projection
T : M -* IP , X H the line through (q,^x)),
1 2
where we think of IP as the space of lines in IP through q. Apply the
Riemann-Hurwitz formula to the map r and obtain
2g - 2 = -2d + E b^,
where the sum is taken over all points x e M of ramification of r , and b
denotes the branch number at x. Note that
103

deg(Tq) = deg(C).
A singular point p e C gives rise to a point x of ramification, where ip{x) = p.
Thus
E b^ = mp - Sp, X e <p~\p).
On the other hand, a nonsingular point p e C with q e T C also gives a point x,
(p{x) = p, of ramification with
b + 1 = # (C,T C) = 2.
X P P
Hence the sum of the branch numbers b over such points is d*. Therefore
2g - 2 = -2d + E (m - s ) + d*. D
The integer d* = deg(C*) clearly depends only on the curve C not the
particular nonsingular model chosen, and is called the class of C. It is the
number of tangent lines of C through a general point of IP \C.
Let Cj, C- be irreducible plane curves of degree > 2. Suppose that a line
L is tangent to both Cj and C . This means that C* and C* meet at L e P^*.
The integer #t(C*,C*) is sometimes called the tangent number. By Bezout's
theorem we find that
E#j^(C*,Cp = cla8s(Cj).class(C2).
Let C be a nonlinear irreducible curve in IP . Given a point p e C we
recall from the preceding section the integers
m = the order of p,
p ^
s = the number of distinct tangent lines at p.
The number s is thus the number of irreducible components of the germ of C at
p. An irreducible component of the germ of C at p is called an irreducible
branch of C at p.
Let Cj,' • ',0 (s = s ) be the irreducible branches of C at p. Put
/i . = # (C.,T C), ti = iti ,,---,/i ).
"^pi " p^ r p r "^p ^'^pl' "^ps'
Observe that
104

a) s = 1 if p is a nonsingular point or a cusp;
b) s =1 and /i = 2 if p is nonsingular and not a flex.
Definition. Let C be an irreducible plane curve of degree d > 2. Then
a) an ordinary m-ple point p 6 C is called a regular m-ple point if none of
the irreducible branches of C at p has p as a flex, i.e., /i = (2,- • ',2);
b) a node p € C is called a flecnode (repectively, biflecnode) if one
(respectively, two) of the branches at p has p as a flex;
c) a line L is said to be m-multitangent to C (m > 2) if L is tangent to C
at m distinct points of C which are not flexes;
d) a cusp p € C is called a simple cusp of multiplicity m if m = m and
^ = #p(C.TpC) = m+1.
If a line L is m-multitangent to C, then L € P is a regular m-ple point
of C . Hence there exist finitely many multitangent lines to C. Also suppose L
€ P is a regular m-ple point of C such that every tangent line at L € C is
not tangent to C* at another point. Then L is a m-multitangent to C. If p 6
C is a flex of order k, i.e., /i = k+2, and further if T C is tangent to C at no
other point, then T C € P * is a simple cusp of multiplicity k+1 of C*. Finally
suppose L = T C € P is a simple cusp of multiplicity k+1 of C* such that
T-C* is tangent to C* at no other point. Then p is a flex of C of order k. In
general a complicated tangent line to C is a complicated singular point of C*.
For the remainder of the section we assume that C is an irreducible plane
curve of degree d > 2 such that
a) C has only simple cusps of multiplicity 2 and regular nodes as singular
points;
b) every flex of C is an ordinary flex;
c) every multitangent is a bitangent, and
d) the tangent line at a singular point or a flex is a tangent line at no
105

other point.
Griffiths [GH] calls such a curve, a curve with traditional singularities.
Put
g = the genus of C,
d = the degree of C,
f = the number of flexes of C,
6 = the number of nodes of C,
K = the number of simple cusps of multiplicity two of C,
b = the number of bitangents to C.
Dualizing we also obtain
g = the genus of C*,
d* = the degree of C*,
d = the class of C*,
K = the number of flexes of C*,
b = the number of nodes of C*,
f = the number of simple cusps of multipicity 2 of C ,
6 ~ the number of bitangents to C*.
Theorem (the classical Pliicker formulae). We have
a) g = Kd-l)(d-2) -6-K,
b) g = i(d*-l)(d*-2) - b - f,
c) d* = d(d-l) -26- 3/«,
d) d = d*(d*-l) - 2b - 3f.
See [GH] pp. 277-280 for a proof.
In the following we give a proof of the genus formula for nonsingular plane
curves which states that for a nonsingular plane curve of degree d its genus is
equal to i(d-l)(d-2). Let C C IP be a nonsingular curve of degree d. Take a
point p € IP \C and a line T ^ P in IP not containing p. We have the
106

projection map
t;C-»T!JJIP\ q€CHL OTeT,
p - ) H p^q )
where L denotes the line through p and q. t is holomorphic and its degree is
d. This means that all but finitely many points of T = IP are covered by r
exactly d times. Let E c T denote this finite set of points. Triangulate T K S
so that every point in E arises as a vertex in the triangulation. We have
V - E + F = 2,
where V is the number of vertices, E, the number of edges, and F, the number of
faces in the triangulation of T. Lifting this triangulation to C via r we obtain
E = d'E many edges, and F = d«F many faces. However, the number of
vertices V is less than d'V in that we must take into account the case where
L is a tangent line to C at q, i.e., q E E. Now the number of tangent lines to
C going through a generic point of IP \C is class(C). So
V = d-V - class(C).
We have
2-2g = V-E + F = d-V- cla8s(C) - d-E + d-F.
Thus 2g = 2 - 2d + class(C) and the genus formula holds if and only if class(C)
equals d(d-l). We now show that class(C) = d(d-l). Let C be defined by the
homogeneous polynomial f(x,y,z) E C[x,y,z]. Dehomogenize f and get F(x,y) =
f(x,y,l), where we assume that the line at infinity {z = 0} is not a tangent line
to C. Without loss of generality p = [1,0,0] € IP^\C and put T = {x = 0} c P^.
Then L (q € C) is a tangent line to C at q iff F (q) = 0. It follows that
class(C) is the number of solutions of the simultaneous system {F = 0, F =0}.
Now
deg(F) = d and deg(F^) = d-1,
and B^zout's theorem takes care of the rest.
Remark. For space curves in general the degree d and genus g are related rather
107

loosely. To give an example, we mention a theorem due to Gruson and Peskine:
Let C c P be a nondegenerate irreducible smooth curve. Further suppose that C
does not lie on any quadric surface. Then
0 < g < gd(d-3) + 1.
Moreover, for every d > 0 and g satisfying the above inequality there is an
irreducible smooth curve in IP of degree d and genus g. For more on this result
the reader may consult an expositiory article by Hartshorne [H2].
§2. Meromorphic Functioiu and Meromorphic Fonni
Observation. Let M. be a compact connected Riemann surface and also let M
be any connected Riemann surface. Suppose we have a nonconstant holomorphic
map
f: Mj -* M2.
Then f is surjective, and consequently M. has to be compact.
Proof. The image f(Mj is open since f is an open mapping. Since f is
continuous f(M) is compact, hence closed in M (M being Hausdorff). d
Let M., M. be Riemann surfaces and consider a holomorphic map
f: Mj -* M^.
About any point p E M. there exists a local holomorphic coordinate z such that
f(z) = z°, for some nonnegative integer n.
To see this first let w be any holomorphic coordinate centered at p so that
f(w) =» E a.w', n € I"*", a M-
i>n ' °
Thus we can write
f(w) = w°h(w),
where h(w) is holomorphic and h(0) ^ 0. Now in a small disc about 0 we can
108

take a single-valued branch of the n-th root of h, say fi. Then f(z) = z° with z
= wh(w). The number n-1, denoted by bip), is called the branch number. If
n > 1, then p (or f(p)) is called a ramification point or a branch point.
Eqnidistribution Property. Let M., M be compact connected Riemann surfaces,
and consider a nonconstant holomorphic map f: M. -* M . Then there exists a
positive integer m such that every q 6 M is taken m times counting multiplicity
by f, i.e.,
E (bj(p) + 1) = m for every q 6 M^,
where the sum is taken over all p € f^ (q).
Proof. For each integer n > 1, put
S^ = {q 6 M^: E (b/p) + 1) > n, p 6 F^q)}.
The local normal form f(z) = z° shows that S is open in M . We will show
that S is also closed in M , Take a convergent sequence (q.) -* q, q. € S ,
Since there are only finitely many points in M that are the images of
ramification points of f, we may assme that b^p) = 0 for any p € f^ (q.) for
every i. Thus f (q.) consists of at least n distinct points. Let p.j,*",p. be n
distinct points in f (q.). Since M is compact, for each j there is a subsequence
of (p..) that converges to a limit p.. Relabeling if necessary we suppose that (p..)
-* p.. (The p.'s need not be distinct.) Easily f(p.) = q, and since f(p.,) = q. we
see that
E (b (p) + 1) > n.
pef-i(q) ^
Thus q € S showing that S is closed. Consequently, each S is either empty or
all of Mg. Let q^ € M^ be any point and put m = E (bj(p) + 1), p € f~ (q^).
Then S = M„, and S ,, = 0 since q^ ^ S ,,. D
m 2' m+1 ^0 '^ m+1
Remark. The number m is called the degree of f, and is equal to f (l), where
f^: H2(Mj,ff) = Zr -. H2(M2,Zr) = 1.
109

Let g. denote the genus of M., i = 1, 2. Put
B = E bj(p), p 6 M.
The integer B is called the total branching number. There is the
Riemann-Hmwitz Formula. 2(gj-l) - 2m(g2-l) = B.
Proof. Put S = {f(p) G M.; hXv) > 0}. S is a finite set and we can triangulate
M- so that every point of S occurs as a vertex. Put
F = the number of triangles,
E = the number of edges,
V = the number of vertices
of this triangulation. Lifting this triangulation to M. via f we obtain a
triangluation of M with F^ = mF , E^ = mE , V^ = mV - B. Now
2 - 2g. = F. - E. + v.,
°i 1 1 i'
and the result follows, o
Hereafter, M = M will denote a compact Riemann surface of genus g
unless otherwise specified.
Definition. A meromorphic function on a compact Riemann surface M is simply
a holomorphic map
f: M -» P^
Choosing a point at infinity in IP , denoted by od, we may identify
P^ = C U {od}.
Given a meromorphic function f on M it is customary to choose od so that
f(M) i {a,}.
Let z denote the usual coordinate on C. Then the map
P^\{0} -► C, Z H I/Z, OD H 0
is a biholomorphism.
Let f be a not identically zero meromorphic function on a compact
Riemann surface M. An immediate consequence of the equidistribution property
110

is that the total number of zeros of f equals the total number of poles taking into
account multiplicity.
Definition. A meromorphic l-form on M is locally given by
f(z)dz,
where z is a local holomorphic coordinate and f(z) is a meromorphic function.
A meromorphic l-form on a Riemann surface is also called an Abelian
differential.
Let f be a meromorphic function on M. Then the total differential df is a
meromorphic l-form. Locally,
df = f'(z)dz.
For example, take
f: P^ = C U {od} -» P^ = C U {od},
Z H 1/z, OD H 0.
Then in P^\{od}
df = -dz/z^
Let u) be an arbitrary meromorphic l-form on a compact Riemann surface
M, and write locally
u) = f(z)dz.
The residue of w at p € M is defined to be
res tJ = res f.
p p
To see that the residue is well-defined just observe that
res w = ^ir w,
P 2m J^ '
where 7 is a small path around p of index 1.
Proposition. Let oj be any meromorphic l-form on a compact Riemann surface
M. Then the total residue
£ res tJ
111

must vanish.
Proof. Triangulate M so that each singularity of tj lies in the interior of a
triangle. Let A.,*",A, be the triangles in this triangulation. Then
E res w = r-T w,
P ^'^ J 7i
where 7. denotes the boundary of A.. Since each edge appears eaxtly twice with
opposite signs the integral must vanish, o
§3. Linear Systems of Divisors
A divisor D on a compact Riemann surface M is a finite formal sum
D = E a.p., a. € ll\{0}, p. € M.
The set of all divisors on M, denoted by Div(M), forms a free Abelian group
under addition. (Div(M) is isomorphic to the free Abelian group on the points of
M.) If every a. > 0, then we say that D is integral (or effective) and write
D > 0.
We have a group homomorphism
deg: Div(M) -» H, deg(D) = E a..
Notation. Ker(deg) = Div°(M).
Let f be a not identically zero meromorphic function on M. It is
convenient to use the sheaf notation and write
f € H°(M. ^)-
Then the divisor of f, denoted by (f), is
(f) = E a.p. - E b.q.,
where the p.'s are the zeros (p. with multiplicity a.^), and the q.'s are the poles
(q. with multiplicity b.) of f. We also write
i% = S a.p., (f)^ = E b.q..
112

By the equidistribution property we have
(f) € Div°(M).
A divisor is called a principal divisor if it is (f) for some f e H°(M, it).
Remark. 1) We will see later that Div (M) is exactly the set of principal
divisors if and only if M is biholomorphic to P .
2) Two divisors are said to be linearly equivalent if they differ by a
principal divisor. We will see later that Div(M)/~ can naturally be identified
with the group H (M, 0*\ called the Picard group of M.
Let w be a nonzero meromorphic 1-form on M. Take a (finite) open cover
(U ) of M and write locally
The divisor of w, denoted by (w), is defined to be the divisor D such that
Define the order of w at p € M to be
ord (jj = ord f,
p p'
where tJ = fdz locally. (If z is another local holomorphic coordinate and if w =
!dz, then ord ! = ord f.)
A divisor D is called a canonical divisor if it of the form D = (w).
Proposition. Let f € H°(M, /). Then
deg(df) = 2g - 2,
where g is the genus of M.
In fact, for any canonical divisor (w), we have
deg(w) = 2g - 2.
Proof of Proposition. Near a pole p 6 M of f, we have the Laurant sense
expansion
f(z) = C_j^Z~'' + • • • + Cq + CjZ + • • • (C_j^ f 0).
Thus
113

df(z) = (-kc_j^z~''~^ + • • • + c_jZ~^ + <^i + ^c^z + . • • )dz.
Near a nonpole q 6 M, we have the Taylor series expansion
f(z) = cz° + c ^,z°+^ + ... (c ^0).
^ ' n n+1 ^ n "^ '
So
df(z) = (nc^z°"^ + ... )dz.
It follows that
deg(df) = E b/q) - E (k(p) + 1),
where p runs over all poles with multiplicity k(p) and q runs over all
ramifications points with branch number n-1 = b-(q) with the proviso that q is
not a pole. Now
deg f = m = E k(p) = the toal number of poles,
and
B = E b/q) + E (k(p) - 1).
The Riemann-Hurwitz formula applied to f now gives
2(g - 1) = -2m + B
= -2 E k(p) + E b/q) + E (k(p) - 1)
= E b/q) - E (k(p) + 1)
= deg(df). D
We now mention an existence theorem for meromorphic 1-forms.
Theorem. Let M be a Riemann surface, compact or not. Specify d, d > 1,
distinct points Pj,*",p. € M. Also let Cj,...,c. be arbitrary nonzero complex
numbers with E c. = 0. Then there exists a meromorphic 1-form w on M such
that u) is holomorphic outside U p.;
ord u = -1, res = c.
Pi Pi »
A proof of the above theorem using the Dirichlet principle can be found in
[FK] Chapter IL
Corollary. Let M be a Riemann surface, possibly noncompact. Then there exists
114

a nonconstant meromorphic function on M.
Proof. Let Pj.Pj.P- be distinct points on M. Let u). be a meromorphic l-form
holomorphic on M\{pj,p } with
ord w, = ord w, = -1,
PI 1 P2 1
res w, = 1, res w, = -1.
PI 1 pa 1
Also let tj be a meromorphic l-form holomorphic on M\{p ,p-} with
ord w„ = ord u)„ = -1,
pa 2 PS 2 '
res W- = 1, res u)„ = -1.
P2 2 ' PS 2
Then the meromorphic function w./w, has a pole at p. and a zero at p.. a
For D € Div(M) we put
L(D) = {f € H°(M, /): (f) + D > 0} U {0}.
Write
D = E a.p, - E b.q. with a.,b. 6 ff"*" and (p.), (q.) all distinct.
Then we see that f € L(D)\{0} if and only if f is holomorphic outside U p. and
ord f > b.: ord f > -a.,
qj " j' Pi - »
It is easy to see that L(D) is a complex vector space. We also have
L(D) = 0 if deg D < 0.
To see this suppose f € L(D)\{0}. Then (f) > -D. So
deg(f) > deg(-D) > 0
which is a contradiction since (f) is a principal divisor.
Observe also that
L(0) = {f: (f) > 0} = {{-. f is entire} = {constant functions} ^ C.
Proposition. Let D > 0 be a divisor on M. Then
dim L(D) < deg(D) + 1.
Proof. Write D = E a.p., a. > 0, (p.) distinct. (If D = 0, then dim L(D) = 1.)
Suppose f 6 L(D). Then about each p. we have the Laurant expansion
115

f= E c.^z^
, ik i'
k=-ai
where z. is a local holomorphic coordinate about p.. Map
*: L(D) -. C^"8(°), f M (c.^), -a. < k < -1.
This map is C-linear with
Ker(*) = {constant functions} = C. a
Notation, dim L(D) = /(D).
Lemma. Suppose D and D. are linearly equivalent. Then
KDj) = ^D^).
Proof. There exists f € H°(M, A*) with D^ - D^ = (f). One routinely verifies
that the following map is a C-linear isomorphism
h € L(Dj) H hf 6 L(D2). a
Ebcample. Let M = IP = C U {od}, and also let (od) denote the point divisor of od.
For d € ff"*" we then have
L(d(a,)) = {f € H°(IP\ /): (f) + d(a,) > 0} U {0}.
We see that for every i with 0 < i < d, the meromorphic function z h z* is in
L(d(oD)). (The function z h z* has exactly one pole of order i at od.) In fact
L(d(oD)) = C-span {1, z, •••, z },
and /(d(oD)) = d+1.
Ebcerdse. Let M = C/L, where L is the integral lattice generated by 1 and i.
Also let p denote the Weierstrass function on M. Show that
L(2(^0))) = C-span {1, p},
where in C -► M is the canonical projection.
For D € Div(M) we put
1(D) = {uK uis di meromorphic 1-form with {J) > D} U {0}.
For D > 0 the set 1(D) consists of all holomorphic 1-forms on M vanishing on
the support of D. In particular,
116

1(0) = H"(M, Kj^) = {holomorphic 1-foras on M}.
Since the degree of any nonzero meromorphic l-form is 2g-2 we have
1(D) = {0} if deg(D) > 2g-l.
It is easy to see that 1(D) is a complex vector space.
Notation, dim 1(D) = i(D).
Lemma. i(D) = /((w) - D) for any meromorphic l-form w -^ 0.
Proof. The following map is a C-^inear isomorphism:
T/ € 1(D) H T]/oj € L((a;) - D). a
The number i(D) depends oidy on the linear equivalence class of D: If D.
and Dg are linearly equivalent, then
i(Dj) = iiu) - Dj) = ^{u) - D^) = i{D^).
We now state the famous
Biemann-Roch Theorem. Let D be any divisor on a compact Biemann surface of
genus g. Then
<D) = deg(D) - g + 1 + i(D).
We will give a proof of the Riemann-Roch theorem in a later section.
Recall that a linear system on a compact Riemann surface M is a
projective subspace A of some |D| = IP(L(D)), or quivalently, a linear system
may be thought of as a linear subspace of L(D) or of H (M, ^(Lj^)) for some D.
The degree of D is also called the degree of A. By the dimension of A we will
mean the projective dimension.
Notation. A = A°, d = deg(A), n = dim(A).
Example. Let M = IP\ and D = d(oD), d € ff"*". Then
|d(»)| = IP(L(d(a,))) ^ P^
since L(d(oD)) = C-span {1, z, •••, z }.
Remark. We will show later that every linear system of degree d on IP is a
subspace of | d(oD) |.
117

Recall that a base point of a linear system A is a point p € M such that p
lies in the support of every divisor in A.
Let A = Aj C IDI be a linear system on M. Via the isomorphism
|D| -»IP(L(D)), D + (f)M {Af: A € C}
identify A with a linear subspace of L(D). Fix a nonzero meromorphic section (
of Lp -♦ M. We then obtain a linear subspace
K^A) = {f.^: f e A c L(D)} c H°(M, 0{h^)).
Then a point p € M is a base point of A C \{0\ i^ ^'^^ only if all sections in
"^(A) vanish at p.
Let D be a divisor on a compact Riemann surface M, and suppose we are
given a base-point-free linear system A = A° c H (M, 0{LJ)). Let {rj , ••, r/ }
be any basis of A c H (M, 0{IjJ)). Then since A has no base points the common
zero locus of holomorphic sections {rj^, •••, rj } is empty, and the holomorphic
map
is well-defined: Each rj. is identified with local data
'i
{r,.^ €h0(U^,^)} such that r/.^ = g^^.„.^,
where {g ^ € H (U n U , 0*)} are transition functions for the line bundle L_ -*
M. Moreover, f. is nondegenerate in the sense that the image f»(M) does not lie
in a lower dimensional projective space: This is so since the set {r/., •••, r/ } is
linearly independent. Moreover, the algebraic degree of f. (M) is equal to d =
deg D: the number of intersections between f. (M) and the hyperplane
{[xq, • • •, XJ 6 IP°; Xj, = 0}
is simply the degree of the divisor of tJq. Choosing another basis of A c H (M,
^(L_)) amounts to changing f. by a projective transformation 6 PGL(n+l, C) =
Aut(IP°).
Conversely, suppose we are given a nondegenerate holomorphic map
118

f: M -» IP°
of algebraic degree d. Then the totality of hyperplane sections of f(M) (a
hyperplane section of S C IP° is the intersection counted with multiplicity of S
with a hyperplane in IP°) is a base-point-free linear system A(f) c |D| on M of
degree d and dimension n. Moreover, choosing a suitable linear basis of K^(A(f)),
where ^ is a nontrivial meromorphic section of L-^ -* M, we can recover f:
A detailed proof of the above result can be found in [Y] pp. 85-87.
Suppose we have n+1 meromorphic functions f., • • •, f € H (M, jt). Put
E = {the polar supports of the f.'s} U {the common zero locus of the f.'s}.
So p € E iff either p is a pole of some f. or all the f.'s vanish at p. We then
have a holomorphic map
f=[fj,, ...,fJ:M\E^IP".
Lemma. The above map f can be uniquely extended to a holomorphic map on
all of M.
Proof. Suppose p 6 E. Put
k = min. ord f..
1 p 1
Then the map
[z\, '", z\]: U C M -» IP°
is a well-defined holomorphic map extending f near p, where z is a local
holomorphic coordinate centered at p. d
Suppose we are given a linear system A C L(D). Let {;/», •••, r/ } be
holomorphic sections of L^ -► M forming a basis of "^(A), where ^ is a
meromorphic section of L_. Put
fi = («erSi = V.J( 6 A C L(D).
Since the map "^ is a linear isomorphism the meromorphic functions f., •••, f
form a basis of A c L(D). By Lemma we then obtain a holomorphic map
119

f=[fo, ...,fJ:M-r
Since rj. = ^f. for every i we see that the two maps f and f. are one and the
same. Hereafter we will use f and f. interchangeably.
Examples
1) Let M = P\ and A = aJ = \d{x)\. Then
f^: P^ -» P'*
is given by
Z H (1, Z, • • • , Z ], CD H [0, 0, • • •, 1].
The complex curve f. (P ) c P is called the rational normal curve.
Remark. We will show later that the set of all nondegenerate holomorphic maps
of degree d from P to P° can be thought of as the complex Grassmannian of
(n+l)-planes in C'*'^\ G(d+1, n+1).
2) Let M be the complex torus given by C modulo the period lattice
J-flpan {wj, w^}, Im(w2/wj) > 0.
We have the holomorphic projection
tt: C -» M,
and the Weierstrass elliptic function ^ on C. The Weierstrass function p on M
satisfies
poir = ^.
Consider the linear system A = |3(7r(0))| on M.
Exercise. L(3(7r(0))) = C-flpan {1, p, p'}.
Proposition. Let A = |3(7r(0))|. Then f.(M) c P is a nonsingular cubic curve.
In fact the linear basis {1, p, p'} of A gives
f^(M) = {[1, X, y] 6 p2: y2 = 4x3 _ g^^ _ g^^^
A A
where g = 60 E (1/w ), g, = 140 E (1/w ) and the sum is taken over all
nonzero periods of ^.
Proof. The Laurent series of ^ and ^' about 0 are:
120

^'(z) = - fa + 2a2Z + 4a3z' + • • • + (2k-2)a^z2^'^ + • • • ,
where a. = (2k-l) E —gj^ and the sum is taken over all nonzero periods of ^. It
follows that
{^'{z)f - i{f>{z)f + 20a2Kz) = -28a3 + ••. .
The right hand side of the above equation represents a holomorphic function near
0 and the left hand side represents an elliptic function with poles at at most the
points in the period lattice. Therefore this function must be a constant, namely
-28a,. This gives the differential equation
The elliptic function ^ projects down to give p on M, and we are done, d
The Canonical Embedding
Let M denote a compact Riemann surface of genus g. The holomorphic
cotangent bundle T*M -» M is also called the canonical bundle and is often
denoted by K^. -♦ M. (For an n-dimensional complex manifold N the canonical
bundle K -* N is defined to be the n-th exterior power of the holomorphic
cotangent bundle.) A holomorphic section of K^. -» M is a holomorphic 1-form
on M. The canonical linear system on M is, by definition, (Z|, where Z is any
canonical divisor. The canonical linear system is well-defined: All canonical
divisors are linearly equivalent to each other since the line bundle asodated to
any canonical divisor is the canonical bundle. The canonical system is often
denoted by |Kj^| or simply |K|, The complete linear system |K„| has
dimension g-1, where g is the genus of M:
1 + dim iKj^l = i(0) = dim hO(M, 0{K^)) = g.
We will see later that if M is nonhypereUiptic, then the holomorphic map
121

is an embedding. In fact, any nondegenerate (irreducible and nonsingular)
algebraic curve c IP^~ of genus g and degree 2g-2 is nonhyperelliptic and is (upto
Aut(IP^~ )) its canonical curve.
§4. The Jacobian Variety and Abel's Theorem
Throughout this section M = M denotes a compact Riemann surface of
genus g. We saw earlier that given any simple closed curve 7 c M there exists a
closed 1-form rj^ unique upto cohomology, such that
I o = I oA T7
J7 Jm ^
for every closed 1-form o on M. The class [r/J is the Poincar6 dual of 7.
Take [a],[b] € H (M,J), where a,b are 1-cycles. We then have
i([a],[b]) = [ T//T/J, = <T/^, -*v^>:
•'M
For embedded cycles a and b, it is not difficult to show that
j »?^ = i(a,b);
'b
and the rest follows from linearity.
Recall that for M = M we have
g
Hjj(M,J) % H2(M,J) % I; Hj(M,J) % J^«.
Let (e ), 1 < o < 2g, be a basis for Hj(M,J). The matrix
J = (J? = (Ke.,ep)
is called the intersection matrix of (e ). A basis (e ) is called a canonical
homology basis if J looks like
where I denotes the g"g identity matrix. The pairing
Hj(M,J) K Hj(M,J) -* I
is completely determined by an intersection matrix.
122

Note that since Hj(M,lf) is free we do not lose any information by looking
at the pairing
Hj(M,IR) K Hj(M,IR) -» R,
where Hj(M,IR) = Hj(M,J) ^^ «•
PTOposition. Let M = M be a Riemann surface of genus g. Given a canonical
homology basis (e ) there exists a unique basis {<p*) of t (M) with
L / - *f, 1 < a./3 < 2g.
Proof. Pick closed 1-forms (r/ ) with the property that
"^ = "^Ar/ , V any closed 1-form.
Put
where x < i < g. We then have
Now {[<p'*]) form a basis of H (M), hence each cohomology class [<p°] contains a
unique harmonic representative. Relabeling, (p^ £ Tl (M). □
The basis {(p*) for 'jlf^(M) is called the dual harmonic basis.
Fix a canonical homology basis (e ) of M. The period map is given by
$: T^(M) - Ir2«, v,H(f ,p).
Proposition. The period map is an isomorphism.
Proof. We assume g > 0. (If g = 0, then there does not exist a harmonic
1-form.) The period map is easily seen to be linear and
'X\m) ^ H^(M) ^ IR^«.
Hence it suffices to show that the kernel of $ is trivial. Suppose (p € Ker($).
So, for every o, we have
123

'k^-l
VAt7,
But then <p = 0 since it is the unique harmonic representative in its class, a
It follows that to give a harmonic l-form on M it is enough to give its 2g
periods relative to a canonical homology basis.
Let (e ) be a canonical homology basis, and {<p°') the dual harmonic basis.
We define T = (Vt) by
Lemma. The matrix P is symmetric and positive definite.
The proof is routine.
Consider the Hodge operator
♦: 'X\m) -» 'x\m).
We want to obtain the matrix representation of * with respect to the basis (y?**).
Let S = (Sp denote this matrix, i.e..
We have
ri = |^'A»^ = l^Asy = l^^sl^v'■" = sj^g,
where 1 < i,j < g. We find that
j+g i+g J » j+g »
Using the matrix notation we have
S = -jr.
Put
a ot • ot
U) = <p + l*<p ,
where {<p'*) is the dual harmonic basis to a canonical homology basis (e ).
Proposition. Each u/* is a holomorphic l-form on M; (u/), 1 < i < g, form a
C-basis of r(K) = H°(M,(7(K)).
Proof. The space of complex valued harmonic 1-forms on M is given by
124

Let f(K) denote the space of antiholomorphic l-forms on M. Now the
proposition follows from the direct sum decomposition
T^(M) = r(K) 9 r(K). D
Proposition. Let M be a compact Riemann surface of genus g, and (e ) a
canonical homology basis. Then there exists a unique basis ((*) of H (M,0(K))
with the property that
J J
(Keep in mind that 1 < ij < g; 1 < a,/3 < 2g.)
Proof. Let A.,«• ',X. be g"g real matrices with
^" L>3 A4j'
where S is the matrix of the Hodge operator relative to (e ). Then
r+u=^:_ji]+i[_;.j«].°
It follows that
We have also
(je i*^) = (^4+iIg' -^3)-
(-A^-ii j; = (r+ij);+8 = -if u/+«,
So
j+g
u
i
We want to change the first g"g block of Q to the identity matrix. Put
'(6 = -*;'-'(i<^-'').
Then
(je/) = (Ig. -A^.-^a;'). d
The above basis (C) is called the dual holomorphic basis. By way of
125

notation
Observation. The matrix D is symmetric with positive definite imaginary part.
Proof. The positive definitejiess follows since A. is positive definite. Knowing
that A. is symmetric it remains to show that A~ A. is also symmetric. Have
'(a;»Aj) - ('a,'a;') = 'Aja;' = a-'aj,
where the last equality is due to the fact that So3 = -I. d
By a complex g-torus we mean the quotient Lie group C^/L, where L is an
integral lattice of translations generated by 2g IR-Unearly independent vectors in
C^. The abelian Jjie group C^/L becomes a complex manifold by requiring that
the projection C^ -♦ C^/L be hplomorphic.
Let M be a Riemann surface of genus g equipped with a canonical
homology basis (e ). The period mfitrix of M is given by
P = (Pi) ^ (|e h
where (^*) is the dual holomorphic basis. Observe that the 2g column vectors
(P 6 C^) are linearly independent over IR. Let <(P )> c C^ denote the integral
lattice generated by them.
Definition. The complex g-torus
J(M) = CV<(P J>
is called the Jacobian variety of M relative to (e ).
Suppose we choose another canonical homology basis, (e ) for M. Then for
some B € GL(2g,ff) we have
e = e-B,
and it is routinly verified that the resulting Jacobian variety is biholomorphic to
the Jacobian variety given relative to (e ). In fact given a canonical homology
basis (e ) and any basis (^*) for H {M,0{K)) we can define the corresponding
126

Jacobian variety to be simply
where L , is the integral lattice generated by the columns of P , = (L ^'): We
then have ^ = A^ for some q € GL(g,C), and
(p y = a!.(p ,)j.
Consequently, the two Jacobians are isomorphic to each other.
A complex g-torus is called an Abelian variety if it can be holomorphically
embedded in a projective space. In the following we give the standard result on
Abelian varieties without proof. For a proof see [GH] pp. 303-304.
Theorem (Riemann Conditions). A complex torus C^/L is an Abelian variety if
and only if there exists a skewsymmetric integral 2g>(2g matrix G such that
P-G~^-*P = 0,
iP'G~ • P is positive definite Hermitian,
where the 2g columns of P generate L.
We can now prove
Theorem. The Jacobian variety J(M) = C^/<(P )> is an Abelian variety.
Proof. We have
P = (Pi) = (je/) = (Ig. «)•
Let G be the 2g"2g matrix whose inverse is the matrix J = V J^\. Then
p.G"^.*p = (i,n).j.*(i,n) = 0.
Also
iP.G"^-*P = i(Q-n).
The result follows since H is symmetric with positive definite imaginary part, d
Let (e ) and (e ) be two canonical homology bases for M. We then have
for some X 6 GL(2g,J), Since e and e are both canonical we must have
127

J = *X.J.X = J,
i.e.,
X € Sp(g,J) = {Y € GL(2g,J): *YJY = J} c SL(2g,2).
Write
for some g»g matrices A,B,C,D. We then find that
(t) ii = (c + Dn).(A + Bn)-\
where (|g 'C) = (1^,11).
Conversely, for any X 6 Sp(g,ff) if fi is defined by (f) then (I ,n) is the
period matrix of M relative to some canonical homology basis (e ).
a'
The Siegel upper half apace is given by
H = {Z € GL(g,C): *Z = Z, Im(Z) > 0 (positive definite)}.
It is a 5g(g+l)-dimen8ional complex manifold. The group Sp(g,2) acts on H by
the prescription given in (f). The upshot of the above discussion is that the map
j: {compact Riemann surfaces of genus g} -* H /Sp(g,2),
M H n (mod Sp(g,J))
is well-defined. We can now state the famous
Torelli Theorem. Compact Riemann surfaces M, M' of genus g are
biholomorphic to each other if and only if j(M) = j(M') € H /Sp(g,J). (It is
known that j is not onto when g > 4.)
For a proof of the Torelli theorem see [GH] pp. 359-362.
Let (Pj,'",p.), (qj,"',q.) € M , the d-fold Cartesian product. We
introduce an equivalence relation on M :
(Pl.-'sPj) ~ (qj.-'-.qd)
if there exists a permutation o" € S. with q. = P^.y
The quotient space M /~ is naturally identified with the set of all integral
128

divisors on M of degree d:
(Pj,...,Pj)(modSj)MEp. 6 Div(M).
By way of notation
Div (M) = {integral divisors of degree d}.
The projection map
m M'* - Div^(M)
makes Div (M) into a d-dimensional compact complex manifold as follows: Let
D = E(p.) 6 Div (M). Also let z, denote a local holomorphic coordinate in U., a
neighborhood of p. in M. Whenever p. = p. we require that (z.,U.) = (z.,U.).
Map 5r(UjK...KUj) C Div^(M) -» C** via
^%» (<^i(Zi(qi)). •••. <^d(Zi(qi))).
where q. € U. and the cr.'s are the elementary symmetric polynomials. We leave
it to the reader to verify that the above map coordinatizes Div (M). Notice that
if the p.'s were all distinct then we could simply take (z .•••,z ) as local
coordinates about D = E(p.).
Let (e ) be a canonical homology basis, and {(^) the dual holomorphic basis
on a Riemann surface M of genus g. As before L denotes the integral
lattice generated by the columns of (P') = ( ^). Fix a point p. € M. Define
V? = V7j: M = Div|(M) -» J(M) = C^L,
called the first jacobi map, by
Vl(p) = (rO(modL).
"'po
To see that <p. is well-defined we let 7 ,7 be two paths joining p. to p. It is
then easily shown that the homology class of the cycle 7^072 is
m'e. + ne ..
1 g+i
for some (m,n) 6 J ^. Consequently
f (C^) - f (0 = ml + nn e L.
"'71 "'72
129

The d-th Jacobi map
(fi^: Div^(M) -» J(M)
is given by
d fPa 1 d rPa _
E(p.)h( E c\ •••, S C«)(modL),
a=l "'po a=l "'po
i.e.,
Vd(^(Pi)) = ^ V'i(Pi)-
In general we can map
ip: Div(M) -» J(M)
via
E(p.) - E(q.) H E v,j(p.) - E v,j(q.).
Note that if D is a divisor of degree 0, then ^D) is defined independent of the
base point p..
Theorem (Abel). Let D,E 6 Div7^(M). Then D is lineary equivalent to E if and
only if ip{D) = ^E) 6 J(M).
We will prove one direction of Abel's theorem: Let D,E be integral
divisors of degree d that are linearly equivalent to each other. So
D - E = (f)
for some meromorphic function f on M. We will show that ^(f)) = 0 € J(M).
Write
(f) = E(p.) - E(q.), 1 < i < d.
Thus
V<(f))= (E0\---.S0«)(modL).
Define a holomorphic map V^ IP -* J(M) by
t = [t(,,ti] H ¥<(tpf-ti)).
The holomorphic cotangent space of J(M) = C^/L at a point is spanned by
dz .•••,dz^, where (z*) are the Euclidean coordinates on C^/L. Each ^*dz' is a
130

global holomorphic l-form on IP , hence must vanish. Consequently ^ is a
constant map, and V = ^[1,0]) = 0. But 0 = V<[1.0]) = ^D-E).
Corollary. For g > 1, the Jacobi map (p = (p.: M -* J(M) is a holomorphic
embedding. In particular, the Jacobi map is a biholomorphism for g = 1.
Proof. The Jacobi map is clearly holomorphic. Since M is comapct and ^M) c
J(M) is Hausdorff it is enough to show that ^ is a one-to-one immersion. Let z
be a local holomorphic coordinate vanishing at p € M and write
^z) = \^\z), ...,/(z))(modL).
Locally
So we can write
^ = T/*dz, rf holomorphic.
<p\z) = f C" + f T7'(z)dz.
PO
Thus
dv^Vdz = T7'(z).
Suppose (fi does not have the maximal rank. Then there exists a point at which
all the holomorphic l-forms vanish. But this is forbidden by the Riemann-Roch
theorem. So rank(^) = 1, and ^ is a surjective map. Let p -^ q € M. If ^p) =
^q), then by Abel's theorem the divisor p-q is principal. Hence there exists a
meromorphic function with a single simple pole which is a contradiction, d
We put
Wj(M) = ip{Diy^{M)) C J(M).
It can be shown that W .(M) is an irreducible analytic subvariety.
Theorem (Jacobi Inversion). Let M be a compact Riemann surface of genus g.
Then for d > g,
Wj(M) = J(M).
Proof. We will prove that W (M) = J(M). The case of d > g is similar.
131

Consider the g-th Jacobi map
V>: Div^(M) -» J(M).
We will show that the rank of the Jacobian of (p is maximal at a point. This
will do since a holomorphic map between equidimensional compact connected
complex manifolds is surjective given that the Jacobian of the map is invertible
at a single point. Let D = E(p.) 6 Div^(M) with the p.'s all distinct. We can
take (z , • • • ,z^) as local coordinates in Div^(M) about D, where z* is a local
coordinate about p. in M. For E = E(z*) near D we have
jfiWE)) = j|,'(E j'V, ..., E fV) ='(CW.
So near D the Jacobian matrix of (p is given by
J(V') = (CVdz^).
Note that changing the local coordinate z' has the effect of multiplying the i-th
column of J(v>) by a nonzero factor. Pick p such that C, (p.) i 0 and
subtracting a suitable multiple of ^ from (^) make C'(Pi) = for i > 1. Now
pick p so that C (P,) ^ ^ ^^^ make ("(Po) = 0 for i > 2. Continuing this
process we arrive at a upper triangular matrix, and we see that the rank of J(^)
at £(p.) is maximal, d
Observe that the g-th Jacobi map (f. Div^(M) -* J(M) is not only
surjective but also injective outside an analytic subvariety: By Abel's theorem
the fibre <p~ (t), t € J(M), is the complete linear system |D|, where D is any
divisor in (p~ (t). Hence (f (t) is a complex projective space. Since Div^(M)
and J(M) are both g-dimensional the fibre over a generic point then must be a
single point.
An immediate consequence of the Jacobi Inversion is the following: any
divisor on M of degree at least the genus of M is linearly equivalent to an
integral divisor.
132

Note that J(P ) is a point. Abel's theorem thus tells us that any two
divisors D,E 6 Div7^(P ) are linearly equivalent to each other so that
Div^(P^)= |d(a,)|.
More explicitly, identify P with C n {od} and put
D = E(p.), E = E(q.)
with p.,q. 6 C. Then
with
D - E = (f)
f=rn(z-p.)/n(z-q.).
So any linear system A° on P is an n-plane in |d(oD)|, and we have
Theorem. Let G°(P ) denote the totality of linear systems of degree d and
dimension n on P^ Then Gj(P^) is naturally identified with PG(d,n), the
Grassmann manifold of n-planes in |d(oD)| = P *.
We now consider a complex torus
M = C/L, L = Jwj ® Jw^.
The space of holomorphic 1-forms H (M,0(K)) is 1-dimensional and is spanned
by dz, where z is the Euclidean coordinate on C/L. Let e denote the homology
class of the cycle represented by the vector w . The period matrix of M relative
to (dz,e ) is
(P J = (je d^ je/^) = («,.-,)•
We thus obtain the explicit identification
J(M) = C/<(PJ>.
Consider the d-th Jacobi map
ip: Div^(M) -» J(M) = M,
fPi
D = E(p.) H E dz (mod L)
i J
We see that
133

ifiD) = pj + ... + p^,
where "+" denotes the addition in the group C/L. Abel's theorem thus says that
two divisors E(p.) and !l(q.) are linearly equivalent to each other if and only if
Pj + • • • + Pj = Qj + • • • + Qj
in the group C/L. The map (p makes Div (M) into a fibre bundle over M with
the standard fibre IP'*"^ (By Abel's theorem v>"^(p) = |D|, D € v>"^(p). Now
|Dj = IP(L(D)) and it is easy to see that dim |D| = d-1.) Put
G°(M) = {linear systems on M of degree d and dimension n}.
The set Gj(M) consists of n-planes in fibres yT (p) = |D|. From this we see
that
G°(M) - M
is a fibre bundle with the standard fibre PG(d-l,n), the Grassmannian of
n-planes in IP
§5. Wderstrass Points and Hyperelliptic Riemann Snifaces
Throughout this section M » M will denote a compact Riemann surface of
genus g.
Definition. Let p € M be an arbitrary point. A positive integer m is called a
gap value at p if there does not exist a meromorphic function f on M with
The point p is called a Weierstrass point if the set of gap values at p is different
from {1, 2, ••., g}.
Examples
1) Take M = P^ = C U {od}. If p € ^\{m}, then put
f(z) = l/(z-p)'", a, M 0.
If p = CD, then put f(z) = z"^. Either way we have (f) = mp, and there are no
134

gaps at any point € IP .
2) Let M be a complex torus, p € M arbitrary. Then there does not exist
a meromorphic functon f with (f) = p: if there were such an f, then f: M -* P
would be a degree 1 holomorphic map, in particular a homeomorphism. So 1 is a
gap value at p. Now consider l(mp), m > 2. Now
i(mp) = 1(Z - mp),
where Z is a canonical divisor, and
deg(Z - mp) = -m < 0.
So i(mp) = 0. By Riemann-Roch
l(mp) = m - 1 + 1 + i(mp) = m.
Consequently there exists a meromorphic function in L(mp)\L((m-l)p), and m is
not a gap value. So at an arbitrary point of a complex torus the set of gap
values is {l}, and there are no Weierstrass points.
Proposition. Let M be a compact Riemann surface of genus g, and p € M an
arbitrary point. Then there does not exist a gap value > 2g.
Proof. (The preceding examples show that the proposition is true for g = 0, 1.)
We will show that
l(mp) - /((m-l)p) = 1 for every m > 2g.
Recall that i(D) = 0 if deg D > 2g-l. It follows that
t(mp) = t((m-l)p) = 0.
On the other hand
i(mp) = i(Z - mp) = (2g-2-m) + 1 - g + ^mp),
t((m-l)p) = l(Z - (m-l)p) = (2g-2-m+l) + 1 - g + l((m-l)p)-
The result follows, a
Proposition. For any p 6 M and m > 1, we have
^mp) - i((m-l)p) = 0, or 1.
Proof. Suppose that l(mp) - l((m-l)p) f 0. Given a meromorphic function f e
135

L(mp) we have the Laurent series expansion
f(z) = a z"™ + ... + a ,z"^ + a- + ... ,
—m —1 u
where z is a local holomorphic coordinate centered at p. Note that
a_ ^ 0 ii and only if f € L(mp)\L((m-l)p).
Recall the C-linear map
*:L(mp)^C'", f H (a_^. a_^_j, ..-.aj.
Suppose fj, fj € L(mp)\L((m-l)p). Then we can find c^, Cg € C so that
*(Cifi + c^g = (0, • •. ),
i.e., Cjfj + c f € L((m-l)p). It follows that one of the f.'s is in the C-span of
the other one and L((m-l)p), and consequently
/(mp) - i((m-l)p) = 1. D
So a positive integer m is a gap value at p € M if and only if
l(mp) - l((m-l)p) = 0;
m is not a gap value at p if and only if
l(mp) - l((m-l)p) = 1.
Now for any point p € M , we have
1 = 1(0) = l(p) < l(2p) < ... < l((2g-l)p) = g < l(2gp) = g+1 < ... .
The following theorem summarizes what we have so far.
Theorem. Let M be a compact Riemann surface of genus g, and p 6 M. Then
there are exactly g gap values at p
m, = 1 < • • • < m < 2g-l.
1 g - ^
Lemma. Let p € M be arbitrary. Then the following conditions are equivalent:
a) p is not a Weierstrass point;
b) l(gp) = 1;
c) i(gp) = 0.
Proof, a) and b) are easily seen to be equivalent: p is a non-Weierstrass point
if and only if the set of gap values is {1, 2, . •., g} if and only if
138

1 = 1(0) = l(p) = l(2p) = ... = l(gp).
b) and c) are equivalent by Riemann-Roch. n
We will show later by Hodge theory that the space of holomorphic sections
of the canonical bundle K.. -* M is g-dimensional. Using the sheaf notation
dim H°(M, OiKj) = g.
Take a C-basis r/ = {r/j, •••, r/ } of H°(M, 0{K^)). Locally we write
T/. = f.(z)dz.
!±i
2
The Wronskian of r/ is defined to be the ^^ - differential given locally by
W(t/, z) = det
, f dz
fjdz,
f;dz^ ... , f'dz
1 ' ' g
f(g-l)dz8
g
To see that W(t/, z) is well-defined globally we let z be another local
holomorphic coordinate and show that W(t/, z) = W(t/, z). We will do this for g
= 2, the general case being totally similar. We have
W(„, z) = (fidz)(fj(dz)2) - (f2dz)(f;(dz)2),
where f' = df./dz. On the other hand
1 i'
f.(z)d2 = f.(z(z)).(dz/dz).dz
so that ?. = f:.(dz/d2). Thus
d?./dz = (dfj/d2)(dz/dz)2 + f..(dVd2^).
It follows that
det
. d!j/dz, dydz
= (dz/dz)"'(det
)■
This shows that
W(t/, z) = W(t/, z).
137

We leave it to the reader to check that W(t/) is not identically zero.
Summarizing we have: The Wronskian of rj is not identically zero holomorphic
fi-ifi - differential on M . To put it another way W(t/) is a global holomorphic
section of the line bundle
As we saw earlier the canonical line bundle is associated to a canonical divisor.
Hence the above line bundle, which can be written as
((g2+g)/2).K^ 6 H^(M, 0*)
under the identification {line bundles over M} ^ H (M, (f), is associated to
((g +g)/2) times a canonical divisor. Thus the degree of this line bundle is
((g'+g)/2)-(2g-2),
which is also the degree of the divisor of the Wronskian of rj. Summarizing
Proposition. The number of distinct zeros of W(t/) € H (M, 0{L)) is at most
g(g+l)(g-l)-
Consider a nontrivial holoniorphic 1-form
w = J; C.T7. € H°(M, d{K)).
Lemma. The form u 6 I(gp) if and only if p is a zero of W(t/).
Proof, u € I(gp) means that p is a zero of u of multiplicity at least g. This is
so if atid only if
E c.f<j)(p) = 0, j = 0, ...,g-l.
The result follows, n
The holomorphic map
f = \L, ..., f ]: M -»IP«-\ T/. = f.dz,
is nothing but the canonical map, and the canonical map is an embedding unless
lA is hypetelliptic. (The hyperelliptic case will be discussed below.) The i-th
derivative map
f(') = #, ...,f(')]:M-IP«-^
V ^1 g ^
138

is called the i-th associated curve of the canonical curve. Thus we may rephrase
the above lemma as stating: p £ M is a Weierstrass point of M if and only if at
p the canonical curve and the g-1 associated curves faii to span P^.
Theorem. Let M be a compact Riemann surface of genus g. Then there are at
most g(g-l)(g+l) many Weierstrass points.
Proof. A point p € M is Weierstrass point if and only if i(gp) ^ 0 if and only if
p is a zero of the Wronskian. d
Lemma. Let p € M and also let {m.: 1 < i < g} be tha gaps at p. Then
ordpW(T/) = E (mj-i).
We leave the proof as an exercise to th^ reader.
Theorem. There are at least 2g+2 many Weierstrass points on a compact
Riemann surface of genus g > 2.
Proof. We have
E (m.-i) = E k - E n. - E i (1 < k < 2g)
= S j (g+1 < j < 2g-l) - E n. (1 < g-1)
< l-g(g-l) - g(g-l) = 6(6-1 )/2,
where n. denotes the i-th nongap at p. So the number of Weierstrass points is
at least (g-l)g(g+l)/(g(g-l)/2) = 2(g+l). d
Corollary. Let # denote the number of Weierstrass points on M . Then
1) # = 2g+2 if and only if at every Weierstrass point the gap set equals
{1.3. .••,2g-l};
2) # = (6-1)6(6+1) if and only if at every Weierstrass point the gaps are
{1,2, •••,g-l, g+1}.
Proof. # = 2g+2 if and only if the order of tl\e WronsJ^ian at every Weierstrass
point is g(g+l)/2. On the other hand ^ = 6(6-l)(6+l) if and only if the order
of the Wronskian at every Weierstrass point is 1. d
139

Definition. A compact Riemann surface of genus at least 2 is called a
hyperelliptic Riemann surface if there exists a meromorphic function f on M with
degree 2 (equivalently, with exactly 2 poles).
Lemma. Let M be a Riemann surface with genus g > 2. Then M is hyperelliptic
if and only if there exists an integral divisor D on M with
deg D = 2, dim L(D) > 2.
Proof. Suppose M is hyperelliptic. Then there is a meromorphic function f with
degree 2. Take D = (f) . Since f 6 L(D)\{constant functions} the dimension of
L(D) is at least two. Conversely, we let D be an integral divisor with degree 2
and /(D) > 2 on an arbitrary compact Riemann surface of genus g > 2. So there
exists a nonconstant meromorphic function g € L(D). Consequently
deg (g)^ = deg g = 1, or 2.
But the degree of g can not be 1 since the genus of M > 1. d
Proposition. Let M be any compact Riemann surface of genus 2. Then M is
hyperelliptic.
Proof. Recall the canonical system |K| on M. We have
dim |K| = g-1 = 1, deg |K| = 2g-2 = 2.
Consider the canonical map
It is not hard to show that this map is base-point-free, hence fij-i is a
holomorphic map of degree 2. (Even without the knowledge that fij,! is
base-point-free we know that it uniquely extends to a holomorphic map.) d
We will see shortly that a generic Riemann surface of genus g > 3 is
nonhyperelliptic.
Let M be a hyperelliptic Riemann surface and also let f be a meromorphic
o
function on M with exactly two poles. Then applying the Riemann-Hurwitz
formula to f we obtain
140

B = 2(g-l) - 4(-l) = 2g+2.
Therefore, we can describe a hyperelliptic M as a two-sheeted cover of P
o
branched at 2g+2 points.
Write IP = C U {od}, and let z denote the inhomogeneous coordinate. Then
PGL(2,C) is identified with Aut(P^) via
A = 0 H v>^, v>^(z) = (az+b)/(cz+d).
Let f be any meromorphic function on a compact Riemann surface M, and
consider
Aof = (af+b)/(cf+d).
If c = 0, then (f) = (Aof) . Otherwise (f) = (Aof) . In any case we have
Theorem. Let M be a hyperelliptic Riemann surface of genus g > 2. Also let f
and I be any two meromorphic functions on M. Then there is a projective
transformation A 6 Aut(IP^) = PGL(2,C) such that ? = Aof. Moreover, the
branch points of f (which coincide with the branch points of ?) are precisely the
Wderstrass points of M.
Proof. Let f and ? be any two meromorphic functions on M with degree 2, and
also let p 6 M be a Weierstrass point. If f(p) = od then (f) = 2p. If f(p) f od,
then (f)^ ~ (l/(W(p)))^ = 2p. Similarly (?)^ ~ 2p, and (f)^ ~ (?)^. Therefore
there exists a meromorphic function h such that multiplication by h gives an
isomorphism
xh: L((!)J -. L((f)J.
{l,f} is a basis of L((f) ) and {1,?} is a basis of L((?) ). It follows that there
are constants a,b,c,d € C such that
f = h(a + bf), 1 = h(c + df),
i.e.,
f = (a + b!)/(c + dl).
141

To prove the latter part of the theorem we fix a meromorphic function f of
degree 2 on M. Let p 6 M be any branch point of f. If f(p) = od then f has a
pole of order 2 at p and no other poles. So 2 is a nongap at p, and p is a
Wderstrass point. In fact we see that the gaps at p are 1,3,« ••,2g-l: the
function v has a pole of order 4 at p, etc. Suppose f(p) i m. Then the function
l/(f-f(p)) has a pole of order 2 at p showing that p is a Weierstrass point with
the gaps {1,3," •,2g-l}. Thus every branch point of f is a gap sequence
1,3,'••,2g-l; since there are 2g+2 branch points already there are no other
Weierstrass points by the corollary at the end of last section, d
Suppose M is a Riemann surface of genus g > 2 with exactly 2g+2 many
Weierstrass points. Then by the corollary at the end of last section the gaps at
every Weierstrass point must be 1,3,'« •,2g-l. Consequently, there exist a
meromorphic function f on M with (f) = 2p, where p is any Weierstrass point.
In particular, M is hyperelliptic. We thus obtain the following characterization
of hyperelliptic Riemann surfaces: A Riemann surface of genus at least two is
hyperelliptic if and only if the number of Weierstrass points on it equals 2g^2.
Recall that a function element (or a power series) determines upon analytic
continuation a multivalued holomorphic function. Let Co'",C- „ (6 ^ 2) be
distinct points in C. Consider the Riemann surface M of the multivalued function
w(z) = V n (z - c.) on H CD.
We think of w as a multivalued function P^ = C U {od} -* C U {od}. The Riemann
surface M is a two-sheeted cover of C U {od} branched at c., • • • ,c- _.
Consequently M is hyperelliptic. Conversely, every hyperelliptic Riemann surface
arises in this fashion. Let M be any hyperelliptic Riemann surface, and also let f
be a meromorphic function with degree 2. Consider the function
w - Ju {i - f(e.)): M -» IP\
142

where the e.'s are the branch points of f. One can then show that the
meromorphic function f is single-valued (see, for example, [FK] pp. 96-98). Now
there exists a unique automorphism of IP sending any three points to 0,1,od. It
follows that the totality of hyperelliptic Riemann surfaces depends locally on 2g-l
parameters. In contrast it can be shown that the set of all Riemann surfaces of
genus g > 2 depends on 3g-3 parameters.
Proposition. Let M be a hyperelliptic Riemann surface of genus g. Also let f be
a meromorphic function of degree 2 on M, and put
w = y n (f - f(e.))-. M -» IP\
where the e.'s are the branch points. Then the set
{f^df/w: 0 < i < g-1}
gives a basis for E°{M,0{K)).
Proof. It is easy to see that the forms {f'df/w} are independent. We will show
that they are holomorphic by showing that each divisor (f*df/w) is positive.
Without loss of generality we assume that for every i, f(e.) f 0,od. Write
(^ = (Pl+Pj) - (P3+P4)-
Then
Also
(df) = (ej+...+e2g^2)-2(P3+P4)-
(w) = (ei+'-'+e^g^^) - (g+l)(P3+P4)-
So
(f'df/w) = (g-i-l)(p3+P4) + iCPj+Pj).
This divisor is positive as long as i < g-1. d
It follows from the above proposition that for a hyperelliptic surface M the
canonical map
143

is given by
ZH[l,f(z),...,f8-^(z)].
So the canonical map is two-to-one and fails to be of maximal rank at the
Weierstrass points.
§6. Projective Embeddings
Let D be a divisor of degree d on a compact Riemann surface of genus g.
Then the Riemann-Roch theorem states that
<D) = deg(D) - g + 1 + i(D).
The following lemma follows from the proposition on the existence of
meromorphic 1-forms we gave earlier.
Lemma. Let Pj,'",p. be distinct points on M = M . Also let c.,"',c. be
arbitrarily chosen complex numbers. Then there exists a meromorphic 1-form w
on M holomorphic on M\{p .•••,p.} and having principal part c/z, at p., where
z. is a local holomorphic corodinate about p.. So near p.,
o;^ = (H + E a.zJ)dz..)
z; j>o
1
We now give a proof of the Riemann-Roch theorem following [GH] pp.
243-245.
Proof of the Riemann-Roch Theorem. We 15rst consider the case of an integral
divisor. Assume that D = p + • • • + p., p.'s all distinct. The case where not
all p.'s are distinct is similar (only notationally more cumbersome) and will be
omitted. Suppose f € L(D). Then df is a meromorphic 1-form on M,
holomorphic on M\supp(D) and a pole of order < 2 at each p.. Moreover, df has
no periods and no residues. Conversely given such a meromorphic 1-form u),
rP
f(p) = a» is well-defined, and f 6 L(D).
•'po
144

Let V denote the complex vector space of all such meromorphic l-forms on M.
Now df = d? iff f-? is a constant. Thus
/(D) = dim(V) + 1.
Let c = (c,,'",c ) 6 C and also let u be a meromorphic 1-form as in Lemma.
Let (e.,e. ), 1 < i < g, be a canonical homology basis of M. Since the period
map is an isomorphism it follows that cj as in the above is uniquely chosen once
we require that
i
0)^=^0, l<\<g.
Ci
Let to denote this normalized meromorphic 1-form. We thus have an
identification
C «—• {the space of Cj 's} via c = (c.,• • • ,c.) -* u).
Consider the map
*: C'* -» C«,
Jeug Jeag
Let (^), 1 < i < g, denote the basis of r(K„) dual to (e.,e. ). We compute
that (using residue calculus)
So the map * is given by the matrix (jMp.))- The number of independent
relations amongst the row vectors of this matrix is the number of linearly
independent holomorphic l-forms vanishing at p. for every i which is equal to
t(D). So
^D) = dim Ker(*) + 1 = d - rank(*) + 1 = d - g + t(D) + 1.
We have proved the theorem for integral divisors. Thus the theorem holds for
divisors of degree > g. For a divisor of degree < g-2 we apply the formula to
Z-D, Z a canonical divisor, and obtain
t(D) = i(Z-D) = 2g-2-d-g + l + i(D).
145

So /(D) = d - g + 1 + t(D). If deg D = g-1 and at the same time neither D
nor Z-D is linearly equivalent to an integral divisor then /(D) = «(D) = 0 and
again the formula holds, d
For a generic integral divisor D = Pj+***+p. of degree d the matrix
(^(p.)) has maximal rank. It follows that
1(D) = 1 for d < g; d-g+1 for d > g
for almost all D. The set of all divisors on M of degree d forms a d-dimensional
compact complex manifold Div (M), and (f) holds for all D 6 Div (M) lying
outside an analytic subvariety.
Definition. By a holomorphic q-differential on a Riemann surface M we mean a
holomorphic section of K-.** -* M. Using the sheaf notation a holomorphic
q-differential on M is an element of
Locally one can be given by
f(z)dz'',
where z is a local holomorphic coordinate and f is holomorphic.
An often useful fact is the following
Observation. Let M be a Riemann surface of genus g > 0. Given any point p 6
M there exists a holomorphic q-differential r/ with ri{p) ^ 0.
Proof. It is enough to show that there is a holomorphic 1-form u) with the said
property. For io{Tp) =^ 0 implies that rj = (J^ does not vanish at p. Recall that m
> 1 is a gap at p iff /(mp) = /((m-l)p). by Riemann-Roch this is so iff
i((m-l)p) = i(mp) + 1.
Now i(mp) = dim I(mp) and
I(mp) = {or. w is a meromorphic 1-form with (w) > mp}
s= {ur. (jj is di holomorphic 1-form with a zero of order at least m at p}.
146

We thus see that m > 1 is a gap at p iff there exists a holomorphic 1-form u
with a zero of order j-1 at p. Now m = 1 is always a gap. So at an arbitrary
point p there is a holomorphic 1-form nonvanishing there, d
Generalizing the identification
L(Z) % H°(M,0(K„)), Z any canonical divisor,
we have identifications
L(qZ) ^ H°(M,(?(K*'')) = H°(MAqK)), q > 1.
Also the vector space of all holomorphic q-diflerentials vanishing at the
points pj,- • '.p is identified with L(qZ - (pj+ h p^)).
Proposition. Let M be a Riemann surface of genus g > 2. Then
i(qZ) = dim L(qZ) = g if q = 1; (2q-l)(g-l) if q > 2.
Proof. We already know that dim L(Z) = dim H°(M,0(K)) = g. Assume q > 2.
By Riemann-Roch
<qZ) = q(2g-2) - g + 1 + i(qZ)
= (2q-l)(g-l) + i(qZ).
But i(qZ) = 0 since deg qZ > 2g-l. d
Pick a basis {t/,,**•)»/»,} of L(qZ). For a local holomorphic coordinate z
we can write r/. = f.(z)dz*'. Then the q-canonical map
f|^2|:M-P^-\N=i(qZ),
is given locally by
Recall that a change of basis of L(qZ) = H°(M,0(qK)) amounts to replacing
f. 21 by Aof. 21 for some A € PGL(N,C) = AutClp'^"^).
Lemma. Let <p = i\ „i be the q-canonical map from a Riemann surface of genus
g > 2. Supppose ^x) = ^y) for some x,y 6 M. Then every holomorphic
q-differential vanishing at x vanishes also at y, i.e.,
L(qZ - x) = L(qZ - x - y).
147

Proof. The reader should have no difficulty in picking a basis {rj.} of L(qZ) such
that
yx) = l.f^2(x) = 0. ....f^^(x) = 0.
where n. = f .dz**, z a local coordinate near x. So
'l XI X' X
<^X) = [1,0,-",0]6PN-^
Since ^y) = ^x) we must have, for any local coordinate z near y,
fyi(y) = ^ M, yy) = 0, •••,fyN(y) = o,
where r?. = f .dz** near y. The rest follows, d
'i yi y •'
We can now establish the following
Theorem. Suppose M is a Riemann surface of genus g > 2. Then the
q-canoiiical map is a holomorphic embedding unless
q = 1 and M is hyperelliptic, or
g = 2 = q.
Proof. Suppose (p = ii j,| is not an embedding. So there exist x -^ y 6 M with
f(x) = f(y). By Lemma we have
KqZ - x) = i(qZ - X - y).
Suppose q = 1. By Riemann-Roch
i(x + y) = 2 - g + 1 + /(Z - X - y).
Now i(Z - X - y) = ^Z - x) = g-1. (L(Z-x) is a hyperplane in L(Z).) So
i(x + y) = 2.
But L(x + y) = {meromorphic functions with at most simple poles at x and y}
and there is a meromorphic function f 6 L(x+y) of degree 2 proving that M has
to be hyperelliptic. Now suppose q > 1. Then
/(qZ - X - y) = q(2g-2) - 2 - g + 1 + /((l-q)Z + (x+y))
= (g-l)(2q-l) - 2 + ^(1^)Z + (x+y)).
And we see that i(qZ-x) = /(qZ-x-y) if and only if
i((l-q)Z + (x+y)) = 1.
148

In particular, in such a case we must have
deg((l-q)Z + (x+y)) > 0.
But this is so if and only if
(l-q)(2g-2) + 2 > 0
which is possible only when q = 2 = g. d
GoioUary. a) Every compact Riemann surface can be holomorphically embedded
in some IP°; b) every nonhyperelliptic Riemann surface of genus 3 can be
holomorphically embedded in IP .
Proof. This follows at once from the preceding theorem together with the earlier
result stating that any complex torus can be embedded in P as a nonsingular
cubic. (It is easy to see that a Riemann surface M^ of genus 0 is biholomorphic
to P : For a generic divisor D of degree 1 on Mj., dim|D| = 1 by Riemann-
Roch and ii-^i gives a biholomorphism Mj. ^ P .) D
Suppose we have a compact Riemann surface of genus g > 2
M C P°, n > 3.
Since the set of all tangent lines to M C P" is 2-dimensional and the set of all
secant lines of M C P° is 3-dimensional we see that a generic point x € P°\M has
the property that the projective line L , y any point € M, is neither tangent to
M at y nor does it intersect M in another point. Without loss of generality we
put X = [1,0,' • ',0]. We can then map t: M h P°~ by
The map t. M -* P°~ is a holomorphic embedding, and since any compact
Riemann surface can be embedded in P° for some n we now have: any compact
Riemann surface can he holomorphically embedded in P .
Remarks
a) Let M be a compact Riemann surface of genus g, and also let Gj(M)
denote the totality of linear systems on M of degree d and of dimension n. It
149

can be shown that Gj(M) is a complex space and that a generic point of Gj(M)
is without fixed point. Put
p = g - (n+l)(g-d+n).
p is called the Brill-Noether number. A theorem due to Kempf, Kleiman and
Laksov says that if d > 1, n > 0, and p > 0, then dim G°(M) > p. In particular,
given that p is nonnegative there exists a nondegenerate holomorphic maps from
M to P" of degree d. For a proof of this theorem and a full discussion of the
Brill-Noether theory see [ACGH] chapters IV and V.
b) A generic (in the sense of moduli) Riemann surface of genus g arises as
a nondegenerate smooth curve in IP° (n > 3) with degree d if and only if the
Brill-Noether number is nonnegative. See [ACGH] p. 216.
c) Given a nondegenerate holomorphic curve f: M -♦ IP°, as we shall see in
Chapter V §2, there arise so called associated curves
f.: M - P^, N = (°+J) - 1, 1 < i < n-1.
Let d. denote the degree of the i-th associated curve f.. An interesting question
to ask is: Given an n-tuple of integers (d,dj,«",d ) does there exist a
nondegenerate holomorphic map f: M -♦ P° whose degree is d and whose
associated degrees are (d.)?
150

Chapter IV. Algebraic Surfaces and the Enriques Classification
Compared with the case of compact Riemann surfaces and algebraic curves
the study of compact complex surfaces and their projective realizations is much
more difficult and lacks cohesion. Whereas a compact oriented topological real
two-manifold admits a unique differentiable structure and a fairly
well-understood moduli of complex structures, even the smoothing theory of
topological 4-manifolds is far from complete. Whereas every compact Riemann
surface is embedded canonically in some projective space, a large class of compact
complex surfaces is not projective. (For example, a generic K-3 surface tan not
be embedded in any projective space.) As a result the theory of projective (and
nonprojective) surfaces tends to concentrate on the study of special surfaces.
An important new notion for varieties of dimension at least 2 is that of a
rational map. The notion of birational equivalence, albeit much weaker than that
of regular equaivalence, turns out to be an extremely fruitful idea; it has far
reaching ramifications in the study of higher dimensional varieties. Early in the
century Enriques and others have already achieved a coarse birational
classification of nongeneral type algebraic surfaces. Our knowledge of general
type surfaces is rather limited, however. The interested reader may consult [Ca]
(and references therein) for a recent discussion on general type surfaces.
Freedman's celebrated theorem says that a compact simply connected
topological manifold is more or less determined by the intersection form. Thus
we look at the intersection pairing
EJiM,I) K H2(M,J) -* I
(and related materials leading up to the Riemann-Roch theorem for line bundles)
in the first section.
In §§2-3 the notions of a rational map and birational invariant are
151

introduced; the Kodaira dimension is defined. In §§4-6 we give a description of
nongeneral type surfaces culminating in the Enriques classification. A somewhat
detailed treatment of K-3 surfaces including the Torelli theorem is given in §7.
In §8 we discuss some results on the posssible Chern numbers of general type
surfaces. (A good treatment on the problem of Chern number geography can be
found in [P2] or [H].) §9 gives a cursory look at singular surfaces and is
essentially unrelated to the rest of the chapter.
Proofs of unproved results in this chapter can be found in the union of
[BFV], [B], and [GH]. We also recommend [BH] as an excellent overall
introduction to the subject matter.
§1. The Intersection Pairing
Let M denote an oriented compact topological 4-manifold. Recall the
intersection pairing
H^CM.J) X H^CM.J) -. I.
It projects down to give a pairing
#: H2(M,2)/Tor x H2(M,2)/Tor -* 2,
where Tor denotes the torsion subgroup.
Pick a 2-basis (e.) for H„(M,2)/Tor % F„(M,2), and put
#„ = #(ei,ep.
Then by Poincar6 duality we must have
det(#) = ±1
as ±1 are the only units in 2.
We have
(H2(M,2)/Tor) • R = H2(M,2) • R = H^CM.R),
and we obtain
# • R: H2(M,R) X H^CM.R) -» R.
152

Now
(H2(M,IR))* = h2(M,IR),
and dualization yields
H^(M,IR) X H^(M,IR) -» R,
which is called the cup product.
If M is a differentiable manifold, then via the de Rham isomorphism the
cup product becomes the wedge product:
A: h2(M) X h2(M) -» R, (A,B) h f ^A^,
where (pyf£ Z^(M) represent the classes A and B respectively.
There is the celebrated
Theorem (M. Freedman, 1981). Given any unimodular symmetric bilinear form
B over the integers of even type (i.e., B(v,v) € 22 for every v 6 2") there exists
a, unique up to homeomorphism, compact simply connected topological
4-manifold whose intersection form is isomorphic to the given bilinear form.
Given a unimodular symmetric bilinear form over the integers of odd type there
correspond up to homeomorphism two compact simply connected topological
4-manifolds and exactly one of these is stably smoothable (i.e., its product with
IR can be given a smooth structure).
Remark. Note that if M is a simply connected n-manifold, then H (M,2) is
necessarily free: the universal coefficient theorem implies
h2(M,2) % FjiM,I) 9 Tj(M,2),
and since M is simply connected H.(M,2) = 0, hence Tj(M,2) = 0. The strong
form of Poincar^ duality states that
H'i(M,2) ^ H^_^(M,2).
In particular.
h2(M,2) % H _„(M,2).
153

So if M is a simply connected 4-manifold, then we also have
TjlM.ff) = 0,
i.e., H (M,Z?) is free.
Hereafter we will let M denote a smooth algebraic surface, i.e., M is a
compact complex 2-manifold that can be holomorphically embedded in some
projective space. For D € Div(M) we have
6(Lp) = Cj(Lp) € e\m,1),
1 4e
where L € H (M,^ ) is the line bundle associated with D and
6: e\m,0*) - e\m,1)
is the connecting homomorphism. We thus obtain a pairing
Div(M) X Div(M) -♦ H,
Send H^(M,J) into H^(M,IR) and identify H^(M,IR) with hJ(M) (the integrality
theorem, proved in Chapter V §1, says that c.(L) is an integral class without
torsion). We thus view
and represent them by closed two forms ^, on M. Then
#(D,,D2) =
Remark. A purely algebraic definition of the pairing
Div(M) X Div(M) -» 1
can be given as follows. Let C and C, be two distinct irreducible curves on a
smooth algebraic surface M, and x € C. n C,, If f is a defining germ of C. in
the local ring 0 , and if g is a defining germ of C„ in ^ , then we put
m^ = dim Oj<{,g>,
where <f,g> denotes the ideal of 0 generated by f and g. By the NuUstellensatz
the ring 0 /<f,g> is a finite dimensional vector space over C. For example, the
154
M

integer m is 1 if and only if <f,g> is the maximal ideal of ^ : in this case the
germs f and g give rise to a local coordinate system near x. We can now define
#(Cj,C2) = S m^,
where the sum is taken over all points of intersection. Extending this definition
linearly over the integers we recover the intersection pairing on Div(M).
We also have the pairing
b.\m,o*) X e\m,o*) - n
given by
(Lj.L^) H #(Cj(Lj),Cj(L2)).
For a divisor D and a line bundle L we define
#(L,D) = #(Cj(L),Cj(Lp)).
We then have
#(L,D) = [ ^L '^ V'd.
•'M
where Vt denotes a closed 2-form on M representing the Chern class c-(L), and
(Pj. is a closed 2-form representing the Poincard dual € H .(M) of the divisor D.
Choose any connection on L -♦ M, and let x € Zj(M) denote its curvature
form. We then have
(ij-Xl = C,(L) 6 H^(M).
We have
where |D| denotes a 2-cycle representing the homology class € H (M,IR) carried
by D.
Remark. Let N be a compact complex manifold of complex dimension n. We
give N the orientation induced from its complex structure. Let S C N be any
k-dimensional analytic subvariety. Recall that the Poincard dual of S is the
On 9V
cohomology class [(pJ € H. (N) satifying the following condition:
155

I o = I o A ^_ for any o e Zj^(M).
Js Jn ^ d
Extending this definition linearly one defines the Poincar^ dual of any divisor on
N. Now let M denote a compact complex 2-manifold. Poincar6 duality gives
the identification
H2(M,IR) = {E^^{M,\R))*,
and
(H2(M,IR))* = e\m,\R).
So a divisor D € Div(M) carries not only a cohomology class (namely, its
Poincare dual € H.(M)) but also a homology class € H,(M,IR),
For the sake of notational simplicity we will write Dj-D, ^"^^t^ad of
#(Cj(Lp ),Cj(Lp )), and D-L instead of #(Cj(L),Cj(L )), and so on. So if
then
^1-^2 = W = ^I'h = ^•^2-
Whenever there is no danger of confusion or when the distinction is not
necessary, we will use the same symbol to denote a line bundle and a divisor
representing it. For example, K will denote the canonical line bundle, or an
arbitrary canonical divisor.
Recall that a type (1,1) form, u, on M is called a positive form if, for local
holomorphic coordinates (z'),
w = i h..(z) dz' A dzJ
with (h..) positive definite Hermitian. A line bundle L -♦ M is said to be
positive if its Chern class can be represented by a positive form. The following
is an easy but useful observation: if L -* M is a positive line bundle and D > 0,
then L-D > 0. To see this just note that L-D is the volume (with multiplicity)
of D with respect to a Hermitian metric arising from the Chern class of L.
156

Let C be a smooth irreducible curve on a smooth algebraic surface M. The
adjunction formula says that
Now
deg(K^) = 2g - 2,
where g denotes the genus of C. And
deg(Kj^ • L^)lc = Kj^.C + L^.C = K-C + CC.
^^v'e have arrived at the genus formula
g = i(K.C + C.C) + 1.
The genus of an arbitrary effective divisor D on M is defined to be
|(K.D + D-D) + 1.
Let D = S a.V. be an effective divisor on a compact complex manifold N,
and S- € H (N,^(L_)). Then tensoring with s. gives an identification between
the meromorphic functions on N with poles of order < a. on V. and H (N,^(L_)).
More generally, if E -♦ N is any holomorphic vector bundle and if ^(E) denotes
the sheaf of germs of its holomorphic sections, then we let ^(E)(D) denote the
sheaf of germs of meromorphic sections of E with poles of order at most a. along
each v.; ^(E)(-D) denotes the sheaf of germs of holomorphic sections of E
vanishing to order at least a. along V.. Again tensoring with s. and s~ gives
identifications
•Sq: <7(E)(D) h 0{E 9 Lj^),
•s~^: <7(E)(-D) « 0{E • L*).
In particular, if C is a smooth analytic hypersurface of N, then the following
sequence of sheaves is exact:
0 -* 0^{E • L*) - 0^{E) i <7p(E|p) -* 0,
where r denotes the restriction map.
Let M be a smooth algebraic surface, and also let C be a smooth
157

irreducible curve on M. We have the exact sequence
0 - Ker(r) - 0^{L^) i 0^{L^) - 0.
Ker(r) is the sheaf of germs of holomorphic sections of L^ vanishing along C,
that is,
Ker(r) = 0„(lj, . lJ) = OJM . €) = 0„.
We thus obtain an exact sequence
(t) ^-*%-* ^m(i^c) -* ^c(i^c) -* 0-
Holomorphic Euler Characteristic. Let E -♦ N be a holomorphic vector bundle
over a compact complex manifold N. Put
hP = dim HP(N,<7(E)).
The holomorphic Euler characteristic of E is defined to be
X(E) = S (-l)PhP.
We usually write x(^m) for x(N x C). The topological Euler characteristic of N is
given by
X(N) = S (-l)P dim HP(N,(|).
In general, if S is any coherent sheaf over N, then we define
X(5) = S (-l)P dim HP(N,5).
Observe that x(E) = x(^(E)) and x(N) = x{%)t where (l^ denotes the constant
sheaf of rationals.
Coming back to our main discussion, from the associated long exact
cohomology sequence of (f) we obtain
where
X(<7c(Lc)) = ^ (-1)^ dim HP(C,<7^(L^)).
The Riemann-Roch theorem for the compact Riemann surface C gives
xiOci^C^) = -g + deg(L^I^) + 1 = -g + C-C + 1.
158

Combining this with the genus formula we obtain
x(Lc) = |(C.C-K^.C) + x(V-
It is not difficult (see [GH] p. 472, for example) to show that the above formula
holds for an arbitrary line bundle L € H (M,^*):
X(L) = |(L-L-K.L) + x(<?).
The preceding formula is called the Riemann-Roch formula for line bundles on a
smooth algebraic surface.
Given a compact complex manifold N we always have
c^(N) = ^j(Kj^),
where as usual c.(N) denotes Cj(TN). To see this first observe that for any
complex vector bundle E -♦ N,
Cj(A'E) = Cj(E).
Therefore
Cj(T*N) = c^(A°T*N) = c^(Kj^).
But Cj(T*N) = -Cj(TN).
The Gauss-Bonnet formula reads
c„(N) = x(N).
For a proof of this we refer the reader to [GH] pp. 409-416.
Let M be a smooth algebraic surface. The following is called Noether's
formula:
For a detailed discussion of this formula see [GH] pp. 600-646.
The following theorem gives a good illustration of the machinary developed
thus far, and is taken from [GH] pp. 487-489.
Theorem. Let M be a smooth algebraic surface with the same Betti numbers as
2 2
IP such that Kj^ is not positive. Then M is biholomorphic to P .
Proof. The Betti numbers of IP are:
159

Recall the exponential sheaf sequence over M
and the associated long exact cohomology sequence
•.. -» H^(M,^) - H^(M,^*) i H^(M,ff) - e\m,0) - •••.
Now by the Hodge decomposition theorem
H^(M,C) = H^'°(M) ® H°'^(M), and h^'° = h°'^
Also H^'°(M) = H^(M,n°) = E^{M,0). It foUows that e\m,0) = 0 and the
connecting homomorphism 6 is injective. Now
e\m,0) = H^(M,n°) = h2'°(M), and
1 = bgCM) > h^'° + h°'2 = 2h2'°
since b2 = h^'^ + h^'° + h°'^ and h^'° = h°'^ It foUows that
So H (M,^) = 0, and S is surjective as well. Thus
e\m,0*) ^ H^(M,ff) ^ ff.
Now
X(<7j^) = S (-l)P dim HP(M,n°) = h°'° - h^'° + h^'" = 1
since h ' =1. Also the topological Euler characteristic of M is
X(M) = S (-l)'b. = 3.
Noether's formula gives
12 = 12x{0) = K-K + x(M).
Hence
K-K = Cj = 9.
The Kodaira embedding theorem tells us that there is a positive line bundle L.
on M. Since E^{M,0*) ^ 1, we can find a generator L of E^{M,0*) such that L^
is a positive integral multiple of L. Since L is positive, so is L. Cj(L) generates
E (M,ff). So L-L = ±1. Using the Kodaira embedding theorem we see that kL
160

is effective for large k. This together with the positivity of L gives
L-(kL) = k(L.L) > 0.
Consequently, L-L = 1. Now
9 = K-K = (mL)-(mL) = m^
So m = -3 and K = -3L. Applying the Riemann-Roch theorem to L we obtain
1 + I (L.L - K-L) = 1 + i (1 - (-3)) = 3 = x(L).
Now x(L) = h° - h^ + h^ where h' = h'(M,<7(L)), and
h^(M,(9(L)) = h\M,0{K 9 4L)) = h\M,0{-4L))
by Kodaira-Serre duality. And by the Kodaira vanishing theorem
h^(M,(7HL)) = h^(M,n^(4L)) = 0.
By Kodaira-Serre duality
h\M,0{L)) = h°(M,(7(K-L)).
But h°(M,<7(K-L)) = h°(M,<7(-4L)) = 0 since ^L is negative. It follows that
h°(M,(7(L)) = 3.
Let D € |L|. If D = Dj+Dg with D. > 0, then
1 = L-L = L-Dj + L-Dg = 2 (note that D ~ L).
consequently, D must be irreducible. If p € D were a singular point, then pick
D' :^ D € |L| with p € D' (we can do this since dim|L| = h°-l = 2). We
would then have 1 = L'L = D-D' > 1. Thus D is also smooth. Now
genus(D) = \ (D-D + D-K) +1 = 0
by the genus formula. So D ^ IP and
L| = the point (= hyperplane) bundle on P .
L| is very ample on P . In particular, the linear system |L| separates points
on each curve D € |L|. Since dim|L| = 2, for any p,q 6 M we can find a curve
D € |L| passing through p and q. Consequently, |L| is base-point-free and
i • M -» P^
is a biholomorphism. □
161

The Degree Revisited
W n
Taking a IP c IP as a generator we may identify
H2^(IP°,ff) = 1.
Via this identification the degree of a k-dimensional variety V c IP° is simply its
underlying homology class. The degree of V is also the number of intersections
I.
between V and a generic IP In the following we give an integral formula
relating the degree and volume. Recall that the KShler form of P" with the
normalized Fubini-Study metric (with the holomorphic sectional curvature 4) is
given by
1=1
where w denotes the puUback of the Maurer-Cartan form of U(n+1) by a local
section of U(n+1) -»!?" = U(n+l)/U(l)xU(n). Observe that IP^ is Riemannian
isometric to the standard two-sphere of radius \. Hence its area is t. More
generally we have
vol(IP") = ^! j *" = ^".
Given a variety i: V «^ P"
since the induced KS.hler form is i $. Now it is not hard to see that the
Poincat^ dual of P°~ is given by
[i, ^'^l € Hf (P'^).
Therefore
deg(vh = #(V^P'^-^) = j i,(i*$)^
We have obtained the
Wirtinger Theorem. deg(V^) = vol(V)-k!.i:i^.
Given a line bundle L on a compact complex manifold M of dimension n
c°(L) € n\ujl).
162

On
The natural orientation on M gives an identification of H (M,ff) with I, and we
define the degree of L to be c°(L) € 1. Suppose L is very ample, i.e., the
complete linear system | L | defines a holomorphic embedding
We then find that
ij^: M - IP^ N = dim|L|
deg(i, (M)) = deg(L).
§2. Rational Maps and the Blow-Up
Definition. A meromorphic function f on a complex manifold is given locally as
the quotient of two holomorphic functions. More precisely, f is given by, for
some open cover {U.},
where g., h. are relatively prime in ^(U.); in ^(U. n U.) we must have
g.h. = g.h..
If g, h are holomorphic functions defined near x € M such that the germs
they define at x are relatively prime, then for y € M sufficiently near x the
germs at y are also relatively prime. A noteworthy feature of a meromorphic
function is that it is not necessarily defined on all of M.
A rational function on IP° is the ratio of two homogeneous polynomials
F, G € C[xjj,...,xj, deg(F) = deg(G), G # 0.
We saw in the first chapter that the field of rational functions on P" is
isomorphic to the quotient field C(x-,"',x). (Just divide F and G by
appropriate powers of x..) A rational function on a variety V c IP° is simply a
rational function on IP° restricted to V.
Proposition. Let V C IP° be a smooth variety. Then a meromorophic function on
V is a rational function, and conversely.
163

Proof. It is easy to see that a rational function is a meromorphic function. We
will show that a meromorphic function f on V is a rational function. By Chow's
theorem the divisors (f). and (f) are expressible as loci of homogeneous
polynomials F(x-,'«',x ) and G(x.,'-',x ). Since (f) is homologous to zero, we
must have deg(f). = deg(f) . Thus F/G is a rational function on P°. Now
(F/G) = (f) implies that f = cF/G for some constant c. a
The totality of meromorphic functions on a complex manifold M naturally
forms a field, denoted by C(M). The transcendence degree of the extension C c
C(M) is called the algebraic dimension of M. Siegel's theorem states that
alg.dim(M) < dim(M).
We know that for a projective manifold the two dimensions coincide.
Definition. A rational map f: M -♦ V c IP° from a complex manifold to a variety
is given by
f:zH[l,fj(z),...,fJz)],
for some global meromorphic functions (f.) on M.
Rather than introducing the notion of a meromorphic map between complex
manifolds we will simply use the two terms "meromorphic map" and "rational
map" interchangeably as the two notions coincide in our setting.
Let S be a Riemann surface and consider a rational map
f: S - IP°, z H [l,f.(z)].
A priori, f is undefined at the poles of the f.'s. At a pole z = 0 replace f by
z H [z\z''f.(z)],
where k is sufficiently large. This way f extends to all of S. Thus
a rational map on a Riemann surface is just a holomorphic map S -* IP°.
On the other hand the following rational map can not be extended to
include the origin in its domain:
f: C^ - f\ (x,y) H [l,y/x] = [x,y].
164

The following theorem gives us a second definition of a rational map.
Theorem, let f: M\W -♦ IP° be a holomorphic map, where W is an analytic
subvariety of codimension at least 2. Then f is a rational map on M.
Conversely, given a rational map f; M -♦ P" there is an analytic subvariety W c
M of codimension at least 2 such that f becomes a holomorphic map on M\W.
Proof. Suppose we are given a holomorphic map f: M\W -» IP° with the
codimension of W at least 2. By Levi's extension theorem (any meromorphic
function on a complex manifold M defined outside an analytic subvariety of
codimension at least 2 extends to a meromorphic function on M) the pullback to
M\W of the inhomogeneous coordinates y. = x./x (i ^ 0) on IP° extend to
meromorphic functions f, on M, So the map [l,f.(z)] is rational. Conversely, let
f: z H [l,f.(z)]
be a rational map. Locally write
with g.,h. € ^(U.) relatively prime. Multiplying by the least common multiple, h,
of the h.'s we may write
f:zH[?^(z),...,?Jz)], ?^ = h.
The local functions (?.) are alll holomorphic, and f is defined away from
n supp(?.) = the common zero locus of the ?.'s.
Note that the ?.'s can not have a common factor. Thus no function vanishing at
a point X € M can divide every ?.. Consequently, n supp(?.) can not contain a
divisor. □
A rational map f: M -» N between varieties is called a birational map if
there exists a rational map g: N -« M such that fog is the identity rational map.
Birational maps are generically one-to-one. An algebraic surface birational to IP
is called a rational surface. Two algebraic surfaces are birational to each other if
and only if their fields of rational functions are isomorphic.
165

Remark. There is also the notion of a rational map between singular projective
varieties. In particular, two possibly singular projective varieties are birational to
each other if and only if their fields of rational functions are (C-) isomorphic to
each other. A deep and fundamental theorem of Hironaka says that any
projective variety is birational to a smooth variety: given any projective variety
V there is a smooth variety N and a birational holomorphic map N -♦ V such
that outside the sigular locus of V this map is a biholomorphism. We have in
mind a birational study of algebraic surfaces; we mostly ignore the singular case.
Basic examples of birational maps are furnished by the blow-up
construction which we explain below.
Let N be an n-dimensional (n > 2) compact complex manifold and p € N.
Take a local holomorphic coordinate system {U, z = (z.)} centered at p and put
U = {(z,0 € U X IP"-1: z € /},
where we think of / as a line in C". There is the projection
T. U -♦ U, (z,/) H z.
Then 7r~^(p) ^ IP"~^ and U\7r~^(p) is biholomorphically identified with U\{p} via
the projection ir. Put
N = (N\{p}) U U, Bp(N) = N/-,
where by decree z € N\{p} is equivalent to (z,/) € U.
Definition, t. B (N) -♦ N is called the blow-up of N at p.
B (N) is naturally a complex manifold making ir holomorphic. We have
Bp(N)\7r-^(p) ^ N\{p}.
E = T~ (p), which is biholomorphic to IP"~ , is called the exceptional submanifold
of IT. Note that E is naturally identified with the tangential directions on M at
p. We leave it to the reader to verify that if N is a smooth algebraic surface,
then so is B (N).
Proposition. Let N be a compact complex manifold, and N = B (N) the
166

blow-up at p. Then
H.(N) = H.(N) e H.(E), i > 0.
Proof. Set
N* = N\{p}, N* = iT^N* = N\E, U* = U\{p}, U* = iT^V* = U\E.
Consider the Mayer-Vietoris sequence of N = N* U U and N = N U U:
H.(U*) -* H.(U) e H.(N*) -* H.(N) -* H.^j(U*),
H.(U*) -* H.(U) e H.(N*) -* H.(N) - H.^j(U*).
Note that U* = U n N*, N = U n N*, and so forth. Now
H.(U) = 0, H.(U) % H.(E),
and TT induces isomorphisms
H.(U*) ^ H.(U*), H.(N*) ^ H.(N*), H.^j(U*) ^ H.^j(U*).
The rest follows, o
If V c N is an analytic subvariety, then the closure of tt" (V)\E in B (N)
is called the proper transform of V.
Let M be a (smooth) algebraic surface and consider the blow-up
in B (M) - M.
Take any irreducible curve passing through p with multiplicity m, and put
C = the proper transform of C = the closure of iT (C\{p}).
Certainly the inverse image divisor tt C is equal to C+kE for some k € 1 . We
will show that, in fact,
/C = C + mE.
Proof. Choose local coordinates x,y in a neighborhood U of p so that the curve y
= 0 is not tangent to any branch of C at p. In the local ring 0 the equation of
C can be written as a formal power series
where the f.'s are homogeneous of degree i (or 0) and f ^0. In a neighborhood
of the point (p,oo) € U c U * IP we can take functions x. and y = ^i/^n ^^o'^l
167

are the homogeneous coordinates on IP ) as local coordinates. Then
/f = f(x„,yx„) = (xoHfJij) + V^^i(i.y) +•••).
and we are done, o
Consider a blow-up B (M) -♦ M of an algebraic surface, and suppose we
have two divisors D, D' on M. Replacing them by linearly equivalent divisors if
necessary we assume that p € M does not lie in any component of D or D'. We
then have
D-D' =r (7r*D).(/D'), E-(7r*D) = 0.
Proposition. Let M be an algebraic surface, and t. B (M) -♦ M the blow-up at
p € M. Then
1) E-E = -1;
2) the assignment
gives an isomorphism
(D,n) H /D + nE
Div(M) ® ff ~ Div(B (M));
3) Kbp(m) = (-*K^) * E.
Proof. Take a curve passing through p with multiplicity 1. Its proper transform
C meets E transversely at a point which corresponds in E to the tangent
direction defined at p by C. So C-E = 1. Now
C = /C - E, /C-E = 0,
and 1) follows. To prove 2) first note that every irreducible curve on B (M)
other than E is the proper transform of its image in M. It follows that the
assignment (D,n) h r D+nE is surjective. To prove that it is one-to-one
suppose there is a divisor D on M such that 7r*D+nE = 0. Then
0 = (/D + nE)-E = n.
It follows that n = 0 and /D = 0. But then D = tt^/D = 0. This proves 2).
To prove 3) choose a meromorphic 2-form a* on M such that w is holomorphic in
168

a neighborhood of p and u{p) ^ 0. Away from E the zeros and poles of r oj
correspond to those of u. Therefore
(/w) = 7r*(w) + kE.
The genus formula shows that k = 1. o
There is the famous
CastelnuovD-Eniiques Ciiteiion. Let M be an algebraic surface and also let C C
M be a smooth rational curve with self-intersection number -1. Then there
exists an algebraic surface M and a holomorphic map t, M -* M making M the
blow-up of M at p with ir {p) = C.
For a proof see [GH] pp. 476-478.
Definition. An algebraic surface is said to be a minimal surface (or a minimal
model) if it does not contain an exceptional curve (i.e., a smooth rational curve
with self-intersection number -1).
A theorem of Zariski states that every algebraic surface can he blown down
to a minimal surface. Moreover, as we shall show later the minimal model is
unique except in the case of ruled surfaces. (A ruled surface is, by definition, an
algebraic surface birational to C * IP , where C is a compact Riemann surface.)
Elimination of Indeterminacy
Let ^: M -♦ V be a rational map from an algebraic surface M to a
projective variety V. Then there exists a surface M', a holomorphic map
t: M' - M
which is the composite of a finite number of blow-ups, and a holomorphic map
f: M' - V
such that (pair =: t (See [B] p. 16 for a proof.) In other words, given a rational
map from an algebraic surface we can replace it by a holomorphic map from a
birationally isomorphic surface. An easy consequence of this is the following
169

Theorem. Let tp: M -♦ M be a birational map between algebraic surfaces.
Then there exists an algebraic surface B and finite composites of blow-ups
r-. B -♦ Mj, t-. B -♦ M
such that tp = "K^iT. .
Low Degree Surfaces in P*
A smooth surface M of degree 2 in IP , called a quadric, is given as the
locus of
E S..X.X. = 0,
where S = (S..) is a nondegenerate complex symmetric matrix. Since all
nondegenerate quadratic forms on C are isomorphic to each other any two
smooth quadrics in IP are projectively isomorphic, i.e., related by an element of
PGL(4,C). There is the Segre map
a: IP^ X IP^ - ip',
(IsQ.Sil.lto.til) H [sQtQ,SQtj,SjtQ,Sjtj].
The map a is a holomorphic embedding and
a(IP^ X ipl) = {xpXj - XjX2 = 0}.
3 11
So any smooth quadric in IP is biholomorphic to IP » IP . Consider the
projection
"^^ ^Vs ~ ^1^2^ "* "* ' K'*"'^J " l^l'^2'^3l-
11 2
The map tt is a birational map, although IP x IP and IP are not biholomorphic.
We have the following classical result: any smooth quadric surface in IP may be
2 2
obtained from IP by blowing up two points Pj.Po ^ "* ^J^d then bloAving down
the proper transform of the line L c IP .
Pipj
3 3
A smooth cubic surface in IP is, by definition, an algebraic surface in IP of
degree 3. Take six points {p.} C IP such that the p.'s do not lie on a conic
curve; no three of them lie on a line. Let t. B(IP ) -♦ P be the composite of the
170

blow-ups at Pj,-",Pg. Also put
E. = the exceptional curve at p.
Consider the complete linear system | C |, where
C = TT 3H — E, — • • • — E-,.
1 0
Then
-1//S
|C| - {it (C) - Ej - • • • - E : C is a plane cubic through p^,- • '.Pg}.
9 1
Proposition, i.pi: B(IP ) -♦ IP is an embedding and its image is a smooth cubic
surface. Moreover, every cubic surface in IP may be obtained by blowing up
some six points satisfying aforementioned conditions and embedding the blow-up
in IP by the proper transform of the linear system of cubic curves passing
through the six points.
We will give a proof of the above proposition later in this chapter.
We will show later that any smooth rational surface in IP is isomorphic to
2
IP , a quadric, or a cubic.
§3. The Kodaira Dimension
let N° be any compact complex manifold. Then
p (N) = h°'° = dim H°(N,n°) = dim H°(N,(7)
is called the geometric genus of N. By duality
p^(N) = dim hO(N,<7(K)),
i.e., it is the number of independent global holomorphic n-forms on N. For a
compact Riemann surface the geometric genus coincides with the usual genus.
The integer
q(N) = h^'° = dim B.\n,0)
is called the irregularity of N. If N is Kahler, then by the Hodge decomposition
theorem we obtain
171

h^'" + h°'l = bj.
And since h ' = h ' we have
q(N) = i bj(N).
The i-th plurigenus is defined to be
P.(N) = dim H°(N,<7(K*')), i > 1.
Note that
Pj(N) = p^(N).
The arithmetic genus of N is defined to be
p (N) = h°'° - h°-^'° + ... + (-l)°-^h^'°.
By duality h'' = h '' and we see that
p,(N) = {-iTixio^) - 1).
Suppose M is a compact complex surface. Then
X{0) = 1 - q(M) + p^(M),
X(M) = E (-l^b. = 2 - 2bj + b^
since b„ = b, and b. = b. = 1.
4 1 0 4
Let f: M -♦ N be a rational map given by a holomorphic map M\V -♦ N,
where M and N are smooth projective varieties, and the codimension of V is at
least 2.
Lemma, f induces a homomorphism f*: H (N,n') -♦ H (M.fl').
Proof. Let <p he & global holomorphic i-form on N. By Hartog's theorem the
puUback f*V'|w\v extends uniquely to a holomorphic i-form on all of M. a
More generally, if E is any contravariant tensor bundle, then we have
f*: e\n,0{%)) -* e\m,0{EJ).
For a birational map f: M -* N the induced map f* is an isomorphism producing
birational invariants.
Corollary. Let N be a smooth projective (hence Kahler) variety. Then the
following quantities are birationally invariant:
172

a) the Hodge numbers h''°(N) (hence p (N), p (N), and xiO^));
b) the irregularity q(N) (hence bJN));
c) the first fundamental group irAU);
d) the plurigenera P.(N), i > 1-
Let M be an algebraic surface. By Noether's formula
12-X«'m) = S + 4
Thus the quantity c.+c. is a birational invariant. However, as we shall see later
the Chern numbers c. = x(M) and c. = K-K are not birational invariants. It
follows that the second Betti number b. is also not a birational invariant.
Let M be an algebraic surface and consider the sequence (P (M)). We
have the following four mutually exclusive possibilities.
1) P (M) = 0 for every n € ff . We then say that the Kodaira dimension
of M, denoted by «(M), is -tD.
Suppose now that P (M) ^ 0 for some n € ff"*":
2) if {P (M)} is bounded, then put «(M) = 0;
3) if {P (M)} is unbounded but P (M) < c-n for some constant c, then we
put k(M) = 1;
4) otherwise, that is, {P (M)/n} is unbounded, then put k(M) = 2. M is
then called a general type surface.
Observation. If k(M) = 0 then P (M) = 0 or 1, n € ff"*".
Proof. If for some m the bundle mK = K had two linearly independent global
holomorphic sections, say 0°,r, then the bundle mnK would have at least n+1
independent sections
We put
n n—1 ^ ^ n—1 n _
a , a 9 T, • ••, (T 9 T , T . 0
R(M) =r ® H°(M,nK„).
n>0 ^
173

For s. € H°(M,iK), s. € H°(M,jK), define s.«s. € H°(M,iK«jK) by
s. • s.(p) = s.(p) • s,(p), p € M.
R(M) becomes an integral domain containing C as a subfield, and
k(M) + 1 = transcendence degree [R(M):C].
Example
1) Let M = IP or any rational surface. Then
q = 1 bj = 0, p = h°(M,K) = 0, and P^ = h°(M,2K) = 0
2
since there are no holomorphic forms on IP . Therefore we conclude that: M is
rational if and only if M is simply connected and all genera vanish.
(Catelnuovo's criterion says that if q = P, = 0, then M is rational.)
2) Let M = IP X C, where C is a smooth elliptic curve. Considering the
2 1
meromorphic 2-form (d^r) , where r. M -♦ IP is the projection, we see that
-2({od} * C) is a canonical divisor. Hence deg(Kj^) = -2. Now deg(Kj.) > 0 iff
there exists a holomorphic 2-form on M. So P. = 0, and P = 0 for every n.
Thus «(M) = -tD. However, M is not rational:
bj(IP^ X C) = bj(IP^).bp(C) + bQ((P^)-bj(C) =r 0-1 + 1-2 = 2, and q = 1.
Rational Maps and Linear Systems without Fixed Part
Let D be a divisor on a compact complex surface M. Recall that the
complete linear system | D | consists of all effective divisors linearly equivalent to
D. Every nontrivial section of L -♦ M defines an element of |D|, and
conversely any element of |D| is the divisor of a section of L -♦ M, defined up
to scalar multiplication. That is to say,
|D| ^ IP(H°(M,(7(Lp)).
A linear system on M is just a projective subspace E < |D|. A curve, or a
positive divisor C c M is called a fixed component of E if every divisor in E
contains C. The fixed part of E is the largest divisor F such that every divisor
174

in E contains F. Put
E- F = {D' - F: D' € E}.
Then E-F, called the variable part of E, is a linear system without fixed part,
and dim(E-F) = dim E. Of course, the system E-F may still have base points.
We have the following basic result: there is a bijective correspondence between
the set of nondegenerate rational maps ^: M -* IP and the set of linear systems
on M without fixed part and of dimension N. The correspondence is constructed
as follows. To the map tp we associate the linear system ^ |H| (the hyperplane
cuts), where | H | is the linear system of hyperplanes in IP . Conversely, let E be
a linear system on M without fixed part and also let E denote the space of
hyperplanes in E. Define a rational map
tp^. M -♦ E* ^ E,
X € M H {divisors in E through x}.
Note that vK^) ^^ defined if and only if x is not a base point.
We can now give a second definition of the Kodaira dimension in terms of
the ampleness of the canonical divisor.
Definition. Let V be a smooth projective variety. Then the Kodaira dimension
of V is the maximum dimension of the images
{V|nK|(^)'="'^'N = dim|nK|:n€ff+}.
If |nK| =0 then ipij^j^\{'V) = 0, and we put dim(0) = -tD. If ac(V) = dim(V)
(note that k(V) < dim(V) always), then we say that V is of general type.
Let C be a smooth curve of genus g. Recall that deg(K) = 2g-2. We
have the following three possibilities:
1) deg(K) < 0 iff g = 0 iff «(C) = -w,
2) deg(K) = 0 iff g = 1 iff «(C) = 0;
3) deg(K) > 0 iff g > 1 iff k(C) = 1.
Proposition. Let M = C. * C, with the C.'s smooth algebraic curves. Then
175

a) if Cj = P^ then k(M) = -cd;
b) if Cj and C^ are both elliptic, then «(M) = 0;
c) if C. is an elliptic curve and genus(C,) > 2, then «(M) = 1;
d) if genus(Cj) > 2 and gen\iB(C^) > 2, then k(M) = 2.
Proof. Let K. denote the canonical bundle on C. and also let ir.: M -» C. denote
1 111
the projection for i = 1,2. We then have
^^1 • ^^2^2 = ^M-
It follows that
E\M,0{nKJ) ^ H°(Cj,<7(nKj)) « E\c^,0{nK^)y
N
Thus the rational map (pi i: M -* P factors as
Cj « Cg -^ P^* X P^' ^ P^,
where f- - fi v \* ^^'^ ^' ([^-lifyil) "(("^yJ) ^^ *^6 Segre embedding. The rest
follows. □
§4. Ruled Sttrfaces
Definition. An algebraic surface M is said to be ruled if it is birationally
equivalent to C » P , where C is a compact Riemann surface. M is said to be
geometrically ruled if there is a holomorphic projection
T. M -* C, C a compact Riemann surface,
whose every fibre is isomorphic to P .
Observe that a rational surface is trivially a ruled surface since it is
2 2 11
birational to P , and since P is birational to IP * P .
Theorem (Noether-Enriques). If M is a geometrically ruled algebraic surface.
176

then it is ruled.
A proof of the above theorem can be found in [B] pp. 25-28.
If E' -♦ M is any holomorphic vector bundle over a complex manifold, then
we define the projectivized bundle P(E') -♦ M to be the fibre bundle over M
whose fibre over any point x € M is the projective space P(E').
Let (U^.^J,
be trivializations for E' -* M. Then the maps <p induce maps
^^:P(E')|^^-*U^.P-l
making P(E') a holomorphic P bundle over M. If
{gab= U, n U, -* GL(r,C)}
are the transition functions for the trivializations (U ytp ), then the transition
functions for P(E') relative to (U ,^ ) are given by the composition
where r. GL(r,C) -♦ PGL(r,C) is the usual projection. If L is any line bundle
over M with transition functions A ,: U n U, -♦ C , then the transition
ab a b '
functions for E' • L are given ^y g'u = ^ u6 u- Consequently, g , = g'u and
P(E') ^ P(E' • L).
Propositioii. Every geometrically ruled surface over a smooth curve C is
isomorphic to P(E') for some rank two holomorphic vector bundle E' -♦ C.
Proof. We will prove the following slightly more general result: any
holomorphic P'~ bundle P over C is of the form P(E') for some holomorphic
vector bundle E' -♦ C. Let
^gab-- U, n U, -* GL(r,C)}
be transition functions for P -♦ C with respect to an open cover U = {U }.
Assuming that U is sufficiently fine we can find liftings
gab^ U^ n U, -* GL(r,C)
177

of g ,. On U n U, n U , set
"ab a b c*
^abc ~ ^ab'^bc'^ca'
Since h . = g ,-g, -g = I we see that
abc "ab ''be "ca
h . : u n u, n u - c*,
abc abc '
i.e., {h , } € Z {11,0 ). From the long exact cohomology sequence of the
exponential sheaf sequence over C and from the fact that H (C,^) = H (C,ff) = 0
we deduce that H (C,^) = 0. So we can write
h. = f ..£. .f
abc ab be ca
for some Cech cochain {f ,: U n U. -♦ C*}. The functions {g ,'^,} then are
the transition functions for a vector bundle E' -♦ C with P = P(E). a
Consider a geometrically ruled surface over an irrational curve
T. IP(E') - C, genus(C) > 1.
If it had an exceptional curve C., then C. could not be a fibre of ir since C.-Cq
= -1. So we must have 'KC.) = C. But this would imply that C is rational
contrary to our hypothesis. Thus we see that a geometrically ruled surface over
an irrational curve is a minimal surface.
Let M be a minimal model of C * P , genus(C) > 1, and also let
(p: M -* C » ?^
be a birational map. Consider the rational map
-K^oip: M -♦ C X P^ -» C.
By the elimination of indeterminacy we can find an algebraic surface M' and
maps
f: M' - C, f. W - M,
where f is a holomorphic map and ^ = ^ o^ o-.-o^ is the composite of n
blow-ups. Moreover, we assume that we chose n to be as small as possible.
Suppose n > 0, and let E be the exceptional curve of the first blow-up f..
Since C is not rational f(E) must be a point. Consider the blow-up
178

Since f is a holomorphic map taking the exceptional curve of the blow-up V, to a
point we can easily find a holomorphic map f: M" -♦ C with f = f'oVi- Now
replacing (M',f) by (M",f') we see that V gets replaced by
contradicting the minimality of n. Thus n = 0, and we obtain a holomorphic
map
TT^oif = tt: M -» C,
where M was an arbitrarily chosen minimal model of C * P . tt is holomorphic
with generic fibre isomorphic to IP . Indeed we have the following
Proposition. Let C be a compact Riemann surface of genus > 1. Then the
minimal models of
M = C X P^
are exactly the geometrically ruled surfaces over C.
For a proof see [B] p. 30.
Digression (Extensions of Line Bundles). Given a line bundle L over a compact
Riemann surface C, a short exact sequence
0 - L' -» E - L - 0, L' € H^(C,<7*),
defines the extension of L by L'. If E ^ L « L', then we say that the extension
is trivial. Note that the extension is trivial if and only if the following sequence
splits:
0 - L » L'"^ - E » L'~^ - ^^ - 0,
i.e., there is a section € H (C, E»L'~ ) which maps onto 1 e E {C,OJ). Using
the exact cohomology sequence
H°(C, E»L'~^) - H°(C, 0^) i H^(C, L»L'~^)
that would mean 6{1) = 0. In fact, 6{1) =: 0 if and only if the extension is
trivial.
179

Coming back to our main discussion we see that classifying geometrically
ruled surfaces over a curve C is the same as classifying the rank two holomorphic
vector bundles on C, up to tensor product with a line bundle. We now briefly
describe the latter classification.
1) Every rank two holomorphic vector bundle on P is decomposable (i.e.,
it is the sum of two line bundles).
2) Every rank two holomorphic vector bundle on an elliptic curve is either
decomposible, or isomorphic to E • L, where L € H (C,^*) and E is one of the
following bundles: 2a) the nontrivial extension of 0„ by 0' 2b) for any p € E,
the nontrivial extension of OJp) by 0„.
3) For every curve C of genus g > 1, the moduli of rank two vector
bundles is at least 3g-3 dimensional.
Let IT. P(E) -♦ C be a geometrically ruled surface over a smooth curve of
genus g, and put F = tt" (x) ^ IP , x € C. Also let K denote the canonical line
bundle on P(E). Then
F-F = 0.
To see this take y ^ x € C such that y is linearly equivalent to x. (Pick a
meromorphic function vanishing at y and a pole at x.) Then F is linearly
equivalent to iT (y). But F-F = F-iT (y) = 0 since iT (y) does not intersect
IT (x) = F. There is the following easy but useful
Observation. If D > 0 and if C is an irreducible curve with C'C > 0 on an
algebraic surface, then D-C > 0.
Proof. Put D = D'+nC for some n > 0 with D' not containing C. Then
D-C = D'-C + nC-C = nC'C > 0. o
Maintaining the above notation we have the following
Proposition, a) F-K = -2; b) e\v{E),0) = 0.
Proof, a) follows from
180

0 = genus(F) = i(F-F + F-K) + 1.
To prove b) suppose H^(P(E),<7) # 0. Then by Hodge duaUty H°(P(E),(9(K)) # 0.
So |K| contains an effective divisor D. But then D-F = -2, contrary to the
preceding observation, o
T c T ^E —» E
i i
P(E) -H C
Let T c iT^E -* P(E) denote the tautological bundle over P(E), i.e.,
T^ = the line given by 1 c E.
The restriction of T to each fibre P(E) ^ P is the universal bundle over P . We
also let Q denote the quotient line bundle on P(E) given by
0 _♦ T -♦ 7r"^(E) -♦ Q -♦ 0.
We then have
Q-F = 1, T-Q = 0,
and the Picard group of P(E) is related to the Picard group of C by
H^(P(E),(7*) ^ /H^(C,<7) ® JQ.
For a proof of the above isomorphism see [B] p. 36.
Proposition. Let M = P(E) -♦ C be a geometrically ruled surface. Then
a) q(M) = genus(C);
b) x(M) = 4 - 4q;
c) K-K = 8 -8q.
Proof. Any holomorphic 1-form ^ on M restricts to zero on the fibres F ^ P .
So ( is the puUback of a 1-form on C. In a neighborhood of any fibre of tt we
may choose a point x and set
181

f(y) = fc
J y
The function f is constant along the fibres and is the puUback of a function g on
an open set in C. Thus ( = 7r*dg and dg is globally defined on C. We have
shown that tt induces an isomorphism
H°(c,n^) ^ H°(M,nji).
(Indeed, H°(M,nj^) ^ H°(C,n^) ® H°(IP\n^j). But H°(IP\n^i) = 0.) But
dim H°(C,n^) = genus(C), and dim H°(M,n^) = q.
The formula in b) follows from
X(M) = x(IP^)-X(C).
Now Noether's formula gives
1 - q = xioj =^2 (K-K + c^).
But c- = 4-4q by Gauss-Bonnet, and c) follows, a
Theorem. Suppose M is a ruled surface (not necessarily geometrically ruled) over
a smooth curve C. Then
q(M) = genus(C), «(M) = -od.
Proof. The irregularity q(M) is a birational invariant. So the first formula
follows from the preceding proposition. To calculate k(M) we may assume that
M = C * P . But then the result follows from a proposition form the preceding
section, o
We can give another proof of the fact that the Kodaira dimension of a
geometrically ruled surface M is -od (without relying on the result that a
geometrically ruled surface is ruled) as follows: P (M) ^ 0 if and only if the
Am
divisor nK is effective if and only if there is a holomorphic section of K -♦ M.
suppose nK > 0. Now F-F = 0 (F ^ P is a fibre and fibres do not meet.) So
F-(any effective divisor) > 0. but this would mean that F-nK > 0 contrary to
the fact that F-K = -2.
182

Finally we mention the famous
Eniiques Theorem. Let M be an algebraic surface with P., = 0. Then M is
ruled.
For a proof see [B] p. 86.
We thus have: an algebraic surface is ruled if and only if its Kodaira
dimension is -tD.
Exercise. Prove the following statements:
a) if M is a minimal surface with c. < 0, then M is ruled;
b) if M is any algebraic surface with c. < 0, then M is ruled.
§5. Rational Surfaces
Definition. An algebraic surface is called a rational surface if it is birationally
equivalent to IP .
We know that any geometrically ruled surface over P is of the form
r. P(E) - P\
where E -♦ P is a holomorphic vector bundle of rank two. Any such E is
decomposible. Hence, for some line bundle L^.Lj € H^(M,<7*),
P(E) = P(Lj e L2).
Now
P(E) ^ P(E • L*) = P(Lj • L* e C),
where C = Cp, denotes the trivial line bundle. Since the Picard group of P is ff,
L • L = nH for some n € ff.
Therefore
P(E) = P(nH ® C).
For n = 0, P(E) ^ P^ x P^ 5) = P(nH ® C) is called the n-th Hirzebruch
surface. From the preceding section we have
183

H^(S^,<?*) = n\\,l) = JQ • JF,
where F is the fibre and Q is the quotient line bundle introduced earlier.
Propositioii. There exists a unique irreducible curve on S with self-4ntersection
number -n for n > 0.
Proof. Let s be the holomorphic section of S -♦ P given by
s(x) = H^.
Put B = s(P^) and write
B = aQ + bF, for some a,b € I.
Now F-B = 1 and a = 1. Q-B = 0 since s*Q = Cpj. Now
0 = B-Q = (Q + bF)-Q = Q-Q + b.
But Q*Q = deg(E) = n and we have
B-B = (Q - nF)-(Q - nF) = n - 2n = -n.
Tp prove the uniqueness of B we let C ^ B be an irreducible curve on S . Set
C = aQ + bF, a,b € 1.
Since C-F > 0 we must have a > 0. Since C-B > 0 and Q-B = 0 we must have
b > 0. Then C-C = a^n + 2ab > 0. o
Corollary. For n ^ m, S and S are not biholomorphic. S is minimal except
for n = 1.
Proof. Now S- ^ P * P and for every irreducible curve C on S , C-C > 0.
2 2
Consider the blow-up M = B (P ), x € P . Projecting away from x we get a
geometrically ruled surface tt: M -♦ P . Now E-E = -1 (E denotes iT (x)) and
we see that M ^ S . So S. is not minimal. The rest follows from the
proposition, o
Proposition. Let M be a minimal rational surface. Then M is biholomorphic to
either P^ or E , n M-
For a proof see [B] pp. 61-62.
There is the famous
184

CastelnuGYQ Theorem. Let M be an algebraic surface with q(M) = P2(^) ~ ^*
Then l\ is a rational surface.
For a proof see [GH] pp. 536-540, or [B] pp. 58-61.
It should be noted that although P„ = 0 implies that p = 0, the converse
^ 6
is false.
The Liiroth Problem
Let V be an n-dimensional projective variety. V is called unirational if
there exists a generically surjective rational map P° -♦ V. To put it another
way, V is unirational if and only if its field of rational functions is contained in a
pure transcendental extension of C. (V is rational if and only if its function field
is a pure transcendental extension of C.)
Theorem (Liiroth). Every unirational curve is rational.
Proof. Suppose C is a unirational curve. So there is a surjective holomorphic
map f: P -♦ C. There exist no nonzero holomorphic forms on C, for the inverse
image of such a form would be a nonzero holomorphic form on P . This shows
that genus(C) = 0. o
As a corollary to Castelnuovo's theorem we can prove the following
Theorem. Every unirational surface is rational.
Proof. Let M be a unirational surface. Then using the elimination of
indeterminacy we can find a surjective holomorphic map M' -♦ M, where M' is a
rational surface. Since q(M') = ?JM') = 0 we then must have q(M) = PjCM)
= 0. Catelnuovo'8 theorem does the rest, o
Surprisingly, it is known that almost all unirational varieties of pure
dimension at least 3 are irrational.
185

The Albanese Variety
Let M be an algebraic surface. The Albanese variety of M is a complex
torus (in fact, an Abelian variety) given by
Alb(M) = (H°(M,n^))*/A,
where A is the lattice of linear functionals obtained by integrating cycles
representing elements in H (M,Zf). The Albanese period map is the holomorphic
map given by
/i: M - Alb(M),
/z(x) = ( Wj, •••, u) (modulo the periods ( w.), a € H,(M,ff)),
where q = q(M), and (w.) a basis for H (M,n ).
We will use the following two facts whose proofs can be found in [B] pp.
66-67.
Fact 1. Let M be an algebraic surface with p (M) = 0 and q(M) > 1. Then
/i(M) c Alb(M) is a curve.
Fact 2. Suppose /i(M) = C is a curve. Then C is a smooth curve of genus q(M)
and the fibres of /x: M -♦ Alb(M) are all connected.
Minimal Model Theorem. Let M be a non-ruled surface. Then there exists a
unique minimal model of M.
We will show that if M and M are non-ruled minimal surfaces birational
to each other, then they are biholomorphic to each other. Let ^ be a birational
map from M to M . As before we have maps
^ = ^ o. • -o^ : M -♦ M , f: M -♦ M,,
where M is an algebraic surface, f is holomorphic, V i^ ^^6 composite of n
blow-ups, and ipoip = f. We further assume that we chose M, ^, and f so that n
is minimal. If n = 0, then we are done. Suppose n ^ 0, and let E c M be the
exceptional curve of the blow-up ^ . We must have f(E) = C, a curve on M ,
186

for otherwise f would factorize as f'o^, with f holomorphic. contradicting the
minimality of n. We now compute C-K.. . (Note that if t. X' -♦ X is the
blow-up of the surface X at a point and C is an irreducible curve on X' such
that t(C') is a curve C, then we would have
Kj^,-C' = {ir*K^ + E)'{ir*K^ - mE) with m = E-C.
Hence
K^,-C' = K^-C + m > Kj^-C
with equality only when C does not meet the exceptional divisor.) We have
K,^ -C < K,^-E = -1
with equality when and only when E dose not meet any of the curves contracted
by f. But in such an event the restriction of f to E is an isomorphism so that C
is a rational curve with K-C = -1, i.e., an exceptional curve, which is
impossible. Thus
Kj^ -C < 2, and by the genus formula C-C > 0.
We must have P (M.) = 0 for every n: if |nK |, n € ff"*", contained a divisor
D we would have D-C > 0 implying that K-C > 0 which is contradictory. We
consider the two cases q(M ) = 0 and qCM.) > 0. Suppose q = 0. Then by
Castelnuovo's theorem M, would have to be rational contrary to the original
assumption that M is non-ruled. Suppose q > 0. Then the Albanese map gives
a surjection
with connected fibres, where B is a smooth curve of genus q. Since C is rational
C is contained in a fibre of /i, say F. Since C*C > 0 we must have
F = rC for some r € ff.
So C-C = 0 and C-K = -2, by the genus formula r =1, genus(F) = 0. But
then by the Noether-Enriques theorem (see below) M would have to be ruled.
For a proof of the following result we refer the reader to [B].
187

Theorem (Noether-Enriques). Let x: M -♦ C be a holomorphic projection from
an algebraic surface to a smooth curve. Suppose there exists x € C such that ir
is smooth over x and iT (x) ^ P . Then M is ruled.
Let M c P° be a nondegenerate (not contained in a lower dimensional
projective subspace) rational, possibly singular, surface. Choosing a birational
map P -♦ M we obtain a nondegenerate rational map
if. V^ -* P°.
It follows that the linear system
{^*H: H a hyperplane in P°}
is of dimension n and without fixed part. Conversely, take a linear system of
dimension n. A, on P without fixed part. Then it gives the rational map
(p-. P -♦ A* ^ P°, X H {divisors in A through x}.
We know that in general we can turn such a rational map into a holomorphic
map by a finite sequence of blow-ups at base points. So if
x: B - P^
is such a blow-up, then the map
f = (poT. B -♦ P°
is a holomorphic map. The holomorphic map f induces a linear system on B
which we denote by f*A.
Take A to be the linear system of all conies on P . Then the dimension of
A is 5, and the rational map
(p^: P^ - P^
is actually a holomorphic embedding. Indeed ^. (P ), is nothing but the Veronese
surface we encountered earlier. Recall also that the degree of ipA^ ) is 4.
Take b, b < 6, distinct points, Pj,'",p, € P . We assume that no three of
them are coUinear; no six lie on a conic. Set d = 9-b, and let
t: B = B^^ - P^
188

be the blow-up at the points p,,'*',p,. Consider
A = A, = the linear system of cubics through p,,* • -.p,.
We then have
Proposition. The dimension of A is d, and the map
f = to(p^. B - P^ - ?^,
is a holomorphic embedding.
f(B) c P is a surface of degree d in P , and is called a del Pezzo surface.
If d = 3, then we recover a smooth cubic surface in P . For d = 4, one can
show that f(B) c P is the complete intersection of two quadric three-folds.
Proof of Proposition. The reader should have no difficulty in verifying that
dim A, = 9 - b = d.
D
We will prove the proposition for b = 6, the other cases being similar. What we
need to check are that the linear system A on P (more precisely, the induced
linear system f*A on B) separates points on B, and tangent vectors on B. Let
i < j < 6, X € B
such that 7r(x) is neither p. nor p.. We will idenitfy B with P away from the
exceptional curves. The general position hypothesis on {p,,'",Pg} implies that
there exists a unique conic Q.. through x and the points p., k ^ i,j. (If x € E
Ij K
the exceptional curve over p., then passing through x means being tangent at p
in the direction corresponding to x.) Similarly there exists a unique conic Q,
through the points p., j # i. Let Q. denote the proper transform of Q.. Then we
see easily that
Q. n Q. = 0, for i # j.
Let L.. denote the line through p. and p., i i j. We now show that the linear
system separates points on B. Take x ^ y € B. Choose i with p. ^ T(x),T(y),
and X ^ Q.. Then
^ij " ^ik "^ W. for Pit i {Pi.PjMx)}.
189

Hence y € Q.. for at most 1 value of j. On the other hand, y € L.. (L., denotes
the proper transform of L,,) for at most 1 value of j. Thus there exists j such
that the cubic
Q.. U L..
passes through x but not y. So the map f: B -♦ P is one-to-one. We now
show that the linear system separates tangent vectors on B. Take
x€PVpi,-.-,Pg}.
The cubics
Q, U h. (L, = the line through p. and x)
do not all have the same tangent at x, so that f is an immersion at x. Now let
X € B- and note that the conies Q,, and (^„. intersect at x with multiplicity 2.
Then the cubics
Q23"V Q24"^3
have different tangents at x. o
§6. The Enriques Classification
W'3 saw that an algebraic surface is ruled if and only if its Kodaira
dimension k is -a>. In this section we give an outline of the classification of
algebraic surfaces with k - 0,1. The classification is originally due to Enriques
and was made precise by Kodaira who also dealt with nonalgebraic compact
complex surfaces.
Hyperelliptic Surfaces
Definition. A compact complex surface M is called a hyperelliptic surface if it is
biholomorphjc to C.xC./G, where the C.'s are elliptic curves and G is a finite
group of translations of C acting on C^ with Cg/G ^ P .
All hyperelliptic surfaces are in fact projective as we shall see below.
190

Let T = C/A be an elliptic curve. Recall that every automorphism of T is
the composite of a translation and a group automorphism. The nontrivial group
automorphisms of T are the symmetry z h -z and the following:
1) for T = T. = C/(ff® ffi), z h ±iz;
2) for T = T = C/(Be Ht), where r' = 1 # r, we have
z H ±TZ, and z h ±r z.
Let C xC./G be a hyperelliptic surface. Since G is a subgroup of Aut(C2)
it can be written as the semidirect product R-A, whete R is a group of
translations and A is a subgroup of the group automorphisms of C,. Since C,/G
li IP , A ^ 0; consequently
A ^ ff/aff, a = 2,3,4,6.
Since G is also a group of translations of C. every element of R is A-invatiant,
and G = RxA, the direct product. The fixed points of A are as follows:
a) for the map z h -z, the points of order 2;
b) for T = T. and A = <z H iz>, the points 0, (l+i)/2;
c) for T = T and A = <z h rz>, the points 0, ±(l-r)/3;
d) for T = T and A = <z h -tz>, the point 0.
Moreover, since R * A is a group of translations of C , it can be generated by
two elements. Hence G is not isomorphic to (C,), * 1/21, where (C,), denotes
the group of points of C- of order 2. We thus obtain
Theorem (Bagnera - de Franchis). Every hyperelliptic surface is one of the
following types:
1) G = 1/21 acts on C^ by symmetry;
2) G = 1/21 e 1/21 acts on C^ by
z H -z, z H z+c, c € (€2)2;
3) C2 = Tj, and G = I/il acts on C2 by z h iz;
4) C- = T., and G = l/Al e B/2B acts on C. by
191

z H iz, z H z + (l+i)/2;
5) Cg = T^, and G = ff/3ff acts via z h tz;
6) Cg = T , and G = ff/3ff • IjZl acts via
z H TZ, z H z + (l-r)/3;
7) Cg = T , and G = 1/61 acts via z h -tz.
Note that in 1) and 2) 2K = 0; in 3) and 4) 4K = 0; in 5) and 6) 3K =
0; in 7) 6K = 0. In all cases 12K = 0. For any hyperelliptic surface we
compute that
h^'° = q = 1, h^'° = 0, h^'^ = 2, cj = 0, c^ = 0.
In particular, we have: every hyperelliptic surface is a minimal algebraic surface
with Kodaira dimension zero.
Let M be a minimal algebraic surface with k = 0. Then
cj = 0, x{0^) > 0, q = 0, 1, or 2.
Proof. The first Chern number c. > 0 since otherwise M would have to be ruled.
2
Suppose c. > 0. Then by the Riemann-Roch theorem
h°(nK) + h°((l-n)K) > x{0) + ^ (nK-nK - nK-K).
So for large n,
h°(nK) + h°((l-n)K) > | cj.
So either h (nK) -♦ an in which case k > 1, or h ((l-n)K) -♦ an in which case k
has to be -od. This proves that c. = 0. Now by Noether's formula
i^'XiO) = X(M) = 2-4q + b2.
Also x{0) = 1-q+P . Thus
8.x(<?) = -2-4Pg + b2>-2-4pg.
But p = P, = 0, or 1. So
g 1
S'x{0) > -«, and x{0) > 0.
The rest follows, o
192

So for a minimal surface with k = 0 we have p = 0,1 and q = 0,1,2. The
two possibilities (p ,q) = (1,1) and (p ,q) = (0,2) will be shown to be impossible,
however.
Case 1: q = 0.
Case la) p = 1. Here we have
X{0) = 1 - q + p^ = 2.
By Riemann-Roch
h°(2K) + hVK) > I (2K.2K - 2K.K) + x{.0) = cj + Cj = 2.
But q = 0 implies that h (2K) = P, = 1 by Catelnuovo's theorem. So -K must
be effective. Since K and -K are both effective we must have K.. trivial.
A compact complex surface M with
bj(M) = 0 and Kj^ = 0
is called a K-3 surface. Such a surface is simply connected (see §7).
Case lb) p = 0, Again P = 0. By Riemann-Roch
h°(3K) + h°(-2K) > x{0) = 1 - q + p = 1.
There exists a nontrivial holomorphic section a of 2K. Suppose h (3K) ^ 0.
Then we would have a nontrivial section r of 3K. Then since P. < 1 we would
have to have
3 2
a - CT for some c € C.
Now if a vanished to order k along any curve C on M, then r would vanish to
order 3k/2. Hence r/a would give a nontrivial section of K, contrary to the
assumption that p = h°(K) = 0. Thus h°(3K) = 0. Consequently h°(-2K) ^ 0.
So 2K and -2K are both effective. Therefore 2K = 0.
A compact complex surface with
q = p^ = 0, 2K = 0 (K M)
is called an Enriques surface. Every Enriques surface is in fact projective.
Case 2: q = 1.
193

We saw that any hyperelliptic surface is minimal, and has Kodaira
dimension zero with irregularity 1 (and p = 0). In fact we have
Theorem. Suppose M is a minimal surface with « = 0, q = 1. Then p = 0 and
M is an hyperelliptic surface.
For a proof see [GH] pp. 585-586, or [B] p. 94. Below we will show that
the geometric genus of a minimal surface M with (K,q) = (0,1) is zero. Since q
is 1 the first fundamental group tt (M) contains a factor of 1. So we can
construct for any m, an m-sheeted unbranched cover r. M' -♦ M. If p =1
then we would have x(^w) = 1- So
So ?JM') =: p (M') > m. Now a section a € H (M',n ) gives rise to a section
since for any p € M and q € tt (p), the fibres of K^ at p and K^, at q are
naturally identified via ir, we can set
%^(p) = ^(Qj) • • • • • aiqj,
where {q,,'",q } = iT (p). If a is not identically zero, then neither is ir^a.
For m > 2 we could find a section a of K ^ -♦ M' vanishing at a point q and
another section r nonzero over 7r(q). Then the images ir a, t t would be two
independent sections of K "^ -♦ M, implying that k(M) > 1. Hence we must have
p (M) = 0. So for a hyperelliptic surface we have the following numerical
invariants:
K = 0, q = 1, p^ = Cj = Cg = 0,
and 12K is always trivial.
Case 3: q =: 2.
Since x(M) > 0 we then must have p =1. Any Abelian surface is
minimal with k = 0, q = 2. Moreover, we have
Theorem. Let M be a minimal surface with k = 0, q = 2. Then M is an
194

Abelian surface. (In particular, K is trivial.)
To prove this theorem we will use the following two lemmas whose proofs
we leave as exercises.
Lenuna 1. Let M be an algebraic surface, C. irreducible curves on M. Set
F = S a,C., a. € H,
and suppose that for every i, F-C. < 0. Also let
D = S b.C. M, b. € 1.
Then
a) D-D < 0;
b) if F is connected and D-D = 0, then
D = rF for some r € (|, and F-C. = 0 for every i.
Corollary. Let r. M -♦ C be a holomorphic projection with connected fibres.
Suppose TT X = S a.C. for some x € C, C. irreducible, a. € ff . Then:
^^ 11 ' 1 ' 1
if D = S bjC., b. 6 1, then D-D < 0 with equality if and only if D = r-7r*x.
Lemma 2. Let f; M, -♦ M_ be a holomorphic projection between surfaces, and
also let {C.} be irreducible curves on M such that f(C.) = x € M for every i.
Then for D = S b.C, b. € 1, we must have D-D < 0.
1 r 1 *
Proof of Theorem. Suppose M is a minimal surface with k = 0, q = 2. So p =
P = 1. Let K be an effective divisor in | K |. Suppose K ^ 0, and write
K- = S a.C., a. € ff"*", C. irreducible.
0 1 i' 1 1
Since c? = K„-K„ = 0 and K„-C. > 0 we must have K.-C. = 0. Hence
10 0 0 1 ~ 0 1
a,C.-C. + S a.C.-C. (i H) = 0-
i 1 1 . J 1 J ^ '^ ->/
J
By Lemma 1 we then have
either C.-G. < 0, and so C.-C. = -2 and C. t> p\
1 1 ' 11 1 - '
or C.-C. = 0 for every i,j.
In
the latter case each C. is a smooth elliptic curve or a rational curve with a
195

double point, and is a connected component of UC Write K = S D , where the
D 's are all connected and effective with D -D. = 0 whenever a # b. The D 's
a aba
are called the connected components of K . Then D -D. = 0 and we have the
following two possibilities: each D is a smooth rational curve, a rational curve
with a double point, or a multiple of one of these; each D is a union of smooth
rational curves. We now consider the Albanese map
/x: M - Alb(M).
We will show that /x is a covering map. (This will do since a cover of an
Abelian variety is again an Abelian variety.) Suppose /x(M) is a curve. Then
/i(M) = B is a smooth curve of genus q = 2 (and the fibres of /x are all
connected). Assume that K. ^ 0. No rational or elliptic curve can map onto B
so that every connected component D of K. is contained in a fibre F of the
fibration /x: M -♦ B. Since D-D - 0 this together with Corollary gives
D = I F with a,b € ff"*".
But then
rb'D = ra-F = /x*(ra'x), where F = /x~^(x).
So h (nD), and with it h (nK), goes to an as n goes to a. This contradicts the
assumption that k = 0. Suppose now that /x(M) is a curve and |K| does not
contain a nonzero effective divisor. (Since dim|K| = 0 we are then saying that
K has to be zero.) So K = 0. Consider a holomorphic covering map
T. B' -*B, where B = /i(M), and deg(7r) > 2.
Set M' = M "gB'. Then M' is connected. Moreover, K , = 0 and x(^m') ^^
also zero. Hence q(M') = 2. But q(M') > genus(B') and by the Hurwitz
formula genus(B') > 3, which is contradictory. Suppose now that /x(M) is
surjective and that K = 0. Let (»?,,%) ^ * ^^^^ o^ H (Alb(M),n ), and set
(J. = /x*r/..
The period map /x is a cover at x € M if and only if the two-form o'jAa', does
196

ip: X^ . ..2
not vanish at x. Since /i is generically a cover WjAw, is not identically zero.
Since K,, is trivial it is then nowhere zero. It remains to deal with the case
M
where n is surjective and K. ^ 0. Let D be a connected component of K . Since
D'D = 0 Lemma 2 shows that /x does not contract D to a point. Thus D can
not be a union of rational curves. So
D = nT, where T is a smooth elliptic curve,
and /i(T) is a smooth elliptic curve T' c Alb(M). Applying a translation, if
necessary, we may assume that T' is a sub-Abelian variety of Alb(M). Consider
the quotient curve Y =: Alb(M)/T' and the holomorphic map
f: M - Y.
The Stdn factorization theorem says that given any proper holomorphic map
between complex manifolds, there exists a complex manifold Z and holomorphic
maps
G: Xj - Z, H: Z - X^
such that tp =s HoG and G is surjective with all of its fibres connected. Thus
there are holomorphic maps
M - Z - Y, f = hog,
g h
such that the fibres of g are all conneted. The curve T is contained in a fibre F
of g. So by Corollary we must have
aT = F, a € ff"*".
Again h (nD) -♦ m as n -♦ m. Thus h (nK) -♦ od as n -♦ od. d
Summarizing what we have so far.
Theorem. Let M be a minimal surface with k = 0. Then M is one of the
fcllowing surfaces:
1) if q = 0, p =0, then 2K = 0 and M is an Enriques surface;
197

2) if q = 0, p =1, then K = 0 and M is a K-3 surface;
3) if q = 1, then p = 0, 12K = 0, and M is a hyperelliptic surface;
4) if q = 2, then p = 1, K = 0, and M is an Abelian surface.
We now briefly look at the case « = 1. But first a
Definition. An algebraic surface M is called an elliptic surface if there exists a
smooth curve C and a holomorphic projection
T.M -* C
whose generic fibre is an elliptic curve.
For any smooth elliptic curve C the ruled surface C * P is trivially
elliptic. Hyperelliptic surfaces are also easily seen to be elliptic. We leave it as
an exercise to the reader to show that every Enriques surface is elliptic.
Theorem. Let M be a minimal surface with « = 1. Then
a) K.K = 0;
b) M is an elliptic surface.
We will need a lemma.
Lemma. Let M be a non-ruled surface. We then have
1) if K-K > 0, then for all sufficiently large n,
^|nK|= M-pN, N = dim|nK|,
is a birational map onto its image;
2) if K-K = 0 and P > 2, then
' n -
K-F = K.(nK -F) = F-F = F.(nK - F) = (nK - F).(nK - F) = 0,
where F denotes the fixed part of the linear system | nK |.
Proof. Take a projective embedding M «^ P and let H denote a hyper plane cut.
Suppose K'K > 0. The Riemann-Roch theorem gives
h°(nK - H) + h^{E + (l-n)K) - a, as n - a,.
We have H-K > 0 as M is non-ruled. Hence
(H + (l-n)K).H < 0 for large n.
198

Thus h°(H + (l-n)K) = 0 for large n. So for large n
h°(nK - H) > 1.
Let E ^ 0 € I nK - HI. The linear system | nK | = | H + E | separates points of
M\E and separates tangents to points of M\E. Thus (p, „• restricted to M\E is
an embedding proving 1). We now suppose that K-K = 0. Now
nK-K = K-F + K.(nK - F) = 0, K-F > 0, K.(nK - F) > 0.
It follows that K-F = K-(nK - F)-= 0. We leave the rest to the reader, a
Proof of Theorem. By Lemma 1) we have K-K < 0. So we must have K-K = 0
for otherwise M would be ruled. Let n be an integer such that P > 2. Let F
denote the fixed part of |nK|. Then (nK - F)-(nK - F) = 0 by Lemma 2). So
the linear system |nK - F| = |nK| - F defines a holomorphic map
f: M - P^,
and its omage is a curve C. Using Stein's theorem we find a surjective
holomorphic map t. M -* B (coming from a factoring M -♦ B -♦ C of f) whose
fibres are all connected. Let T be a fibre of ir. Then K-T = 0 and genu8(T) is
one so that the smooth fibres of ir are elliptic curves, o
§7. K-3 Surfiices
Definition. A connected compact complex two-manifold (not necessarily
algebraic) is called a K-3 surface if b = 0, K = 0.
We note that a K-3 surface M is simply connected: We have
H^(M,ff) ^ Fj ® Tq, and Hj(M,ff) ^ (H^(M,ff))*,
where F. denotes the free part of H (M,ff) and T is the torsion subgroup of
Eq{M,T[). Now T^ = 0 since Hjj(M,ff) ^ 1. But b^ = 0, hence Hj(M,ff) = 0. It
now follows that ^-(M) = 0 since EAM,1) is the abelianization of irAM).
Let M be a K-3 surface. Its canonical bundle K., is then trivial, hence
199

Consequently M carries a nowhere vanishing holomorphic two-form $ which is
determined uniquely up to multiplication by a scalar X € C*.
Let M be a K-3 surface. Then it is easy to see that
dim e\m,0^) = 0;
From the Hodge theorem
Now H^(M,C) = 0 since b^ = 0. Thus
0 = H^'°(M) • H°'^(M).
But h^'° = h°'\ and h^'° = 0. Moreover, we have
Proposition. CgCM) = 24, x{0^) = 2, E^{M,1l) ^ ff".
Proof. By Kodaira-Serre duality.
So h\0{K^)) = h\0j = 1. Now
X{0J = S {-iTi^^Oj = 1-0 + 1 = 2.
Also, by Gauss-Bonnet
Cg = X(M) = S (-1)\ = 1 - 0 + bj - bj + 1 = 2 + b2,
since b, = b = 0 by Poincar6 duality. Noether's formula gives
12.x(M) = K-K + Cj = 0 + (2+b2).
Thus Cg = 24 and b^ = 22. Now
H2(M,ff) % F^ ® Tj.
But Tj = 0 since Hj(M,ff) = 0. So HgCM,?) ^ (H^(M,ff))* is torsion free, o
It is worth pointing out that for a simply connected space the homology
and cohomology groups are always free and H*(X,ff) ^ (H (X,ff))* for every a.
Let M be a K-3 surface and fix a nonvanishing holomorphic two-form $ =
$ . Since d$ = 0, $ defines a cohomology class
[* J € h2(M,C).
200

Fix h basis (e,) of Hj(M,I) and identify Hj(M,if) = l''^. Put
•'ei
(Remember that we do not distinguish e. with a cycle representing it. Likewise
we often confuse a cocycle with a closed 2-form in its cohomology class when
integrating.) We call /i., the i-th period. Let (^) be closed two-forms
representing the dual basis to (e.) so that
M M
Note that /i. € C, and each ^ is a real-valued closed two-form on M representing
an integral homology class.
The intersection pairing
#: e\m,1) X e\m,1) -* I,
2 2
makes H (M,ff) c H'!(M) into a lattice. (A lattice is a free ff-module endowed
'M
^ ''\Jf\ in+n a IniiifP i
with a bilinear symmetric integer-valued form.) We write
L(M) = (H^(M,i7),#) = the lattice of M.
Since any two K-3 surfaces are homeomorphic to each other it follows that
their lattices are isometric to each other.
Theorem. Let M be a K-3 surface, and L(M) its lattice. Then
a) L(M) is unimodular, i.e., the determinant of any integral matrix
representing # is ±1;
b) L(M) is even, i.e., #(x,x) € 21 for any x € H^(M,ff);
c) 8ignature(L(M)) = (3,19).
Proof, a) follows from Poincar6 duality. Suppose x = [D] € H (M,ff) for some
divisor D. By Riemann-Roch we have
(t) x(<?(Ld)) = Kd-d-k.d) + x(V
Suppose now that x € H (M,ff) does not come from a divisor. Even then x is
201

still the class of a smooth C*-bundle on M and we have the smooth version of
Riemann-Roch (see [Hirl]) stating that the right hand side of (f) is an integer.
Finally c) is a consequence of the index theorem of Hirzebruch:
K-K -2C2 = 3(n"*" -n~),
where n"*" (resp., n~) denotes the number of positive (resp., negative) eigenvalues
of #. Just observe the left hand side is -48. a
We have from lattice theory
Theorem. A unimodular indefinite lattice is determined up to isometry by the
rank, index and pairity. The same holds for a definite unimodular lattice of rank
at most 8.
Combining the preceding two theorems we see that a K-3 lattice is
isomorphic to
Ih $ 111 6 IH $ (-E) * (-E),
where IH (called the hyperbolic plane) = (ff ,#J with the matrix of #. relative to
the canonical basis equal to r' ; E = {1 ,^ ), where the matrix of #, relative
to the canonical basis is the Cartah mattix of the Lie algebra e..
The Weak Torelli Theorem. Let M and N be K-3 surfaces. Then M and N are
biholomorphic to each other if and only if there is ah isometry
<p: L(M) - L(N)
such that for some A € C
where (^„ denotes the complexificatioh of (p, and [4>] denotes the cohomology
dass.
For proofs of the two Torelli theorems in this section see [Ba].
Thus a K-3 surface is uniquely determined (up to biholomorphism) by the
line
202

C$ € P^^ = IP(H^(M,C)).
The two-form $ must satisfy
(a) #($,$) = 0,
(b) #($,$) > 0,
where we extend # complex-linearly. The equation (a) defines a smooth
21
hyperquadric Q in P . The equation (b) then defines an open subset
n c Q.
Theorem. The moduli of K-3 surfaces is 0. (0 is a 20-dimensional irreducible
quasi-projective variety.)
Proof. This follows from the weak Torelli theorem combined with so called the
surjectivity theorem: For each p € fl there exists a K-3 surface M and an
isomorphism H (M,C) -♦ C sending the line Cfl to p. a
Remark. A generic member of fl does not represent an algebraic surface- In
fact, the moduli of algebraic K-3 surfaces is 19-dimensional.
We have
h2'0(M,C) = C[$^], H°'2(M,C) = Cl$J
and H^'^(M) c H^(M,C) is the orthogonal complement to H^'° ® H°'^. Moreover,
Re$, Im$ € H (M,IR) span a 2-dimensional positive definite subspace in I<(M)«IR.
Therefore, on
H^'^(M,IR) = H^'^(M) n H^(M,IR)
the intersection pairing has signature (1,19).
Let N be any compact complex manifold. Recall that a homology class in
H.. (N,7) is said to be analytic if it is a rational linear combination of classes of
analytic subvarieties. An even dimensional integral cohomology class is said to
be analytic if its Poincar6 dual is. (The famous Hodge conjecture states that:
on N c P° every rational cohomology class of type (k,k) is analytic.) We now
state the
203

Lebchetz Theorem on (I,l)-Cla88e8. For N c 0*° a submanifold, every
cohomology class a € H ' (N) n H (N,ff) is analytic. In fact, a is associated to
some divisor D on N.
For a proof of the above theorem see [GH] pp. 163-164.
Coming back to our main discussion we have, for a K--3 surface M,
H^(M,<7*) = Pic(M) ^ e\m,1) {] H^'^(M).
Proof. We know that S: Pic(M) -♦ H (M,ff) is an injection. By the preceding
discussion we also have
e\m,1) n H^'^(M) '=^ Div(M)/~ % Pic(M).
Finally given any line bundle, its Chern form is a (1,1) form. So L defines a
class in H^'^(M). o
Thus Pic(M) = H^(M,<7*) is a sublattice of L(M) = (H^(M,ff),#). It is
called the Picard lattice. The Picard number, denoted by p(M), is defined to be
the rank of the Picard lattice. Since rank(H ' (M)) = 20 we have
0 < p{M) < 20.
A K-3 surface with /? = 20 is often called an exceptional K-3 surface.
It can be shown that if M is algebraic, then the signature of Pic(M) is
(l,p-l). See, for example, [BPV].
Let C be a smooth irreducible curve on a K-3 surface M. We write, as
usual, L^ to denote the line bundle associated to the divisor C. We also identify
5 imacrA in TT^M 77^ iinHor iho rnnnprt.incr man A CJn T^
A
Pic(M) its image in H (M,ff) under the connecting map 6. So L^ € Pic(M) c
2
H (M,ff). By adjunction we obtain
Now deg(K_) = 2g-2, where g denotes the genus of C. On the other hand
= C.K„ + C.L^ = C.K„ + C.C.
204

Since K,. is trivial, K-C = 0, and
2g -2 = C-C.
Therefore, we must have
either C-C > 0, or C-C = -2.
If C-C = -2 then C is a rational curve, and is called a nodal curve.
Take any d € e\m,1) with d^ = d-d = -2. Put
r^: L(M) - L(M), x h x + (x-d)d,
where as usual x-d denotes the intersection pairing. The map r. gives the
reflection about the hyperplane d"*". The map r. is an isometry since for any x,
rj(x) = x^ + 2(x-d)2 + {x'dfd^ = x^.
We also note that r .(d) = -d, and that r. restricted to the hyperplane d"^ is the
identity on d"^.
Proposition. The isometry r. can not be induced by an automorphism of M.
Proof, r. takes some effective divisor to a noneffective divisor: r. takes d to -d
d d
and since d = -2 either d or -d is effective, but not both, d
Notice that if d € Pic(M) c H^(M,ff) with d^ = -2, then ^j^-d = 0. In
particular, the map r. is an isometry preserving the period.
Put
A^ = {d € Pic(M): d^ = -2},
AjJ = {d € Aj^: d is effective},
D = {nodal curves}.
So Dj^ c AjJ c Aj^ c Pic(M), and Dj^ i^ AjJ since not every divisor in AjJ can
be represented by an irreducible curve.
On H^'^(M,IR) the signature of # is (1,19) and the set
{h € H^'^(M,IR): #(h,h) = h-h > 0}
breaks up into two half cones ±C-.. Without loss of generality we assume that
the Kahler classes lie in C., = +C-,. Put
205

Cjt is called the Kahler cone of M. In fact, if h € Cjt then h-d > 0 for every
effective divisor d. We can now state the
Strong Torelli Theorem. Let M, N be K-3 surfaces. Suppose we have an
isometry <p: L(M) -♦ L(N) with V'(*m) ^ ^^m" "^^^^ ^ ^^ induced by a unique
biholomorphism N -♦ M if and only if ^T>(Cjt) C C^.
Take a homogeneous quartic polynomial
f(x) = f(Xjj,--.,X3) € C[Xjj,--.,Xj]
and consider the projective variety S c P defined by it. It is easy to see that
S is smooth iff {di/dx.) # 0.
We suppose that S is smooth. We then have
Proposition. bJS) = 0 and Kg = 0.
Proof. Embed P c P and think of S as a smooth hyperplane section in P .
Then by the Lefschetz hyperplane section theorem
i*: Hl(P^^) -* H^(S,^)
is an isomorphism, where i: S «^ P . Thus b.(S) = 0. By adjunction
Ks = (Kp, . 1^)1 J.
Now deg(L ) = 4. On the other hand, deg(Kp,) = -4. Thus Kg ^ Og. □
Consequently, a smooth quartic surface in P is a K-3 surface. A generic
smooth quartic does not contain any P , i.e., a nodal curve. However, there are
special quartics containing up to 64 lines. (Segre showed that this bound is
sharp. The maximum number of lines on a smooth quintic in P is unknown!)
Example
Let f(x) = xj + x| - Xg - Xj. Then S(f) is called the Fermat quartic.
The Picard number of S(f) is 20. Also as we shall see below S(f) contains
exactly 48 lines. Let f., f, be homogeneous quartic polynomials in two variables
and put
206

f(x) = fi(xQ,Xj) - f2(x2,X3).
Also put
Lj = the line {x^ = x^ = 0}, L^ =: the line {x. = x = 0},
{p,.' • •.P4} = the zeros of f^, {q.,* • -.q^} = the zeros of fg.
Note that S(f) is smooth iff |{p.,q.}| = 8. Thus for a smooth S(f) we get
16 lines L(p.q.) c S.
The tanget plane T (S) cuts out on S, 4 lines L(p.q,), In fact, any line L c S
other than the 16 lines {L(p.q.)} is skew with L and L,, since L can not contain
any p.. We thus have a (bi-)holomorphic map
^l' \ -* Lg. P " (pVL) " ^2'
where (p V L) denotes the plane through p and L.
Observation. ir^: {Pi,---,p4} ■=► {qi,---,q4}-
Proof. L meets the plane p.V L in a point on one of the lines L(p.q,). a
Given any biholomorphism ir: L -♦ L identifying {p.} with {q.} we may
identify the variables (xq,xJ with (x2,x,), hence identify f. with f,. We see that
the four lines
then lie on S, and they determine ir. Thus the number of lines on S equals
16 + 4n,
where n is the number of it's as in the above.
Exerdse. Show that n = 0,4,8, or 12.
So the number of lines lying on a smooth quartic c P defined by
f(xQ,Xj) - f2(x2,X3)
is 16, 32, 48, or 64.
Kummer Sur&ces
Let A be an Abelian surface. Choosing an origin makes A into a group
207

C /A. Let a be the involution of A given by z h -«, The fixed points of a are
the points of order 2 of the group A which is group-isomorphic to (R/ff) . So
there are 16 points of order 2, p,,'*',p,g. Let
T = 7r,«o. •'OTT,: A -♦ A
10 1
be the composite of the blow-ups at the points p-.-'-.p-g. The involution a of
A then extends to an involution a of A.
Proposition. M = A/{1,5-}, called the Kummer surface of A, is an algebraic K-3
surface.
See [B] pp. 99-100 for a proof.
Normal Projective K-3 Surfoces
Suppose M is a K-3 surface embedded in P° and further suppose that M is
normal, i.e., the embedding is given by a complete linear system. Let C = M-H
be a generic hyperplane cut of M and consider the exact sequence
Since K = K-j is trivial we see from the adjunction formula that
So the linear system cut out on C by hyperplane sections of M c P° is a
subsystem of the canonical system on C. Moreover, we know that the linear
system of hyperplane cuts of M is the complete system |H (M,^ (C))|, and
h\M,OJ = q(M) = 0,
so that H°(P°,<7(H)) maps onto H°(M,<7(C)) which maps onto H°(C,n^). We see
that the hyperplanes in P° thus cut out the complete canonical system on C, i.e.,
C c H = ? is a canonical curve. C has genus n and degree 2n-2. In
particular, a normal K-3 surface in P° has degree 2n-2. We can give another
proof of this fact as follows: since C = H-M is positive
h\M,0{C)) = hl(M,n^(C)) = 0,
208

and similarly h (M,^(C)) = 0 by the Kodaira embedding theorem. The
Riemann-Roch theorem gives
n + 1 = hO(M,<7(C)) = i CC + x{OJ = | deg(M) + 2,
and thus deg(M) = 2n-2.
We already saw that smooth quartics in P are normal K-3 surfaces. The
linear system of quartics in P has dimension 34, and PGL(4,C) = Aut(P ) has
dimension 15. Thus the family of quartics in P is 19-dimensional. We will now
consider smooth sextic surfaces in P . If C is a hyperplane section of M, then
the system of quadrics in P cuts out on M a linear system of dimension at most
h°(M,<7(2C)) - 1 = i 2C.2C + 2 - 1 = 13.
Since the linear system of quadrics in P is 14-dimensional, M, then, must lie on
a quadric hypersurface Q C P . Now
h°(M,<7(3C)) = ^ + 2 = 29, h°(P^<7(3H)) = 5.6.7/6 = 35.
Thus M must lie on a 5-dimensional family of cubics in P . But the system of
cubics containing the quadric Q has dimension
h°(P^<7(H)) -1 = 4.
So M must lie on a cubic Q' not containing Q. Since Q is irreducible, Q' must
meet Q along a surface of degree 6 or less, and hence exactly along M. Thus we
have: a smooth sextic M c P is a K-3 surface that is the complete intersection
of a quadric and a cubic. Conversely, if M = Q n Q' is such a smooth complete
intersection, then applying the Lefschetz hyperplane section theorem twice we get
q(M) = 0, and by adjunction
^M = (Kq + Q)Im = (V + Q' + Q)Im = (-^H + 3H + 2H)|^ = 0.
Hence M is a K-3 surface. A sextic in P is determined by choosing a quadric Q
(14-dimensional) and then a cubic Q' in the 35 - 5 - 1 = 29-dimensional family
of cubics in P modulo those containing Q. Now dim(PGL(5,C)) = 24. Thus the
family of sextic K-3's in P is again 14 + 29 - 24 = 19-dimensional.
209

We will now look at Enriques surfaces briefly. Let M be an Enriques
surface. Then, by definition,
Since 2K is trivial.
Pg = q = 0, x{OJ = 1-
c? = 0.
By Rieraann-Roch
X(M) = 12.
The Hodge numbers are
h°'° = h2.2 = 1,
h^'O = h°'l = h2.0 = h°'2 = h^'l = hl'2 ^ 0^
h^'^ = 10.
Theorem. Every Enriques surface is the quotient of a K-3 surface by a fixed-
point-free involution, and vice versa.
For a proof see [GH] pp. 514-544,
§8. General Type Surfaces
Theorem. Let M be a minimal surface. Then the following conditions are
equivalent:
a) « = 2, i.e., M is of general type;
b) K'K > 0, and M is irrational;
c) for sufficiently large n, v^i j, i restricted to its image is birational.
Proof. We already saw that a) is equivalent to c). Let M be a surface with K
= 0. Then for every n € ff"*",
(nK -F).(nK - F) = 0,
where F denotes the fixed part of |nK|. This implies that dim(^| Tr|(M)) < 1.
We thus have shown that c) implies b). (Recall that if M is minimal with K <
210

0, then M is ruled.) Finally an irrational surface with K positive is non-ruled.
The lemma at the end of §6 says that the image of v^i jfi, for large n, has to be
2-dimensional. a
It can be shown that for a general type surface the linear system |nK| is,
in fact, base-point-free for n < 4.
The following summarizes the known restriction on the Chern numbers of a
minimal general type surface.
Theorem. Let M be a minimal general type surface. Then
1) c^ > 0, Cj > 0;
2) cj + Cj = 0 (mod 12);
3a) Cj > Cg/S - 36/5, if cj is even;
3b) Cj > Cg/S - 6, if c^ is odd;
4) cj < 30^.
Proofs of the above results can be found in [BPV] Chapter 7.
Property 2) is an immediate consequence of Noether's formula, and as such
it holds for any algebraic surface. Property 3) can be rewritten as Noether's
inequality
P,<|cJ + 2.
Property 4) is the famous Miyaoka-Yau inequality.
A theorem of Kodaira-Chow states that if the algebraic dimension of a
compact complex surface M is 2, then M has to be algebraic. Now the Kodaira
dimension can not exceed the algebraic dimension, hence, there are no non-
algebraic surfaces with k = 2.
Definition. A smooth complete intersection of type (d ,-",d ), d. 6 ff"*"\{l}, in
IP° is a smooth surface M which is the transverse intersection of n-2 algebraic
hyper surfaces (not necessarily smooth) of degree d,,- • •,d ,.
Suppose M is a smooth complete intersection. We then have (for a proof
211

see [BPV] p. 137)
/c(M) = -OD, if M is of type (2), (3), or (2,2);
«(M) = 0, if M is of type (4), (2,3), or (2,2,2);
k(M) = 2 otherwise.
A smooth complete intersection of type (2) is a quadric in P . All smooth
A J
intersections of type (3) in P and of type (2,2) in P are also rational, and they
can be obtained from P by blowing up 6 and 5 points respectively. Observe
that a smooth algebraic surface in P is of general type if and only if its degree is
at least 5.
Observation. Let N = B (M) -♦ M be the blow-up. Then
C2(N) = C2(M) + 1; cJ(N) = c2(M) - 1.
Proof. Firstly note that the second formula follows from the first since
I2.x(<7) = c2+C2
is a birational invariant. To prove the first formula recall
H.(N) = H.(M) • H.(E), i > 0,
where E ^ P^ is the exceptional curve. Now bj(E) = 0 and \{E) = b2(E) = 1.
So
C2(N) = x(N) = £ (-1)'H.(N)
= S (-1)'H.(M) + b2(E) = C2(M) + 1. D
Corollary. Let M c P be an algebraic surface not of general type. Also let d
denote the degree of M. We then have the following possibilities:
1) d = 1, M ^ p2, c^ = 9, C2 = 3;
2) d = 2, M ^ P^ X p\ c^ = 8, C2 = 4;
3) d = 3, M ^ the blow-up of P^ at 6 points, c^ = 3, C2 = 9;
4) d = 4, M is a K-3 surface, c = 0, c- = 24.
We can summarize what we know about the Chern numbers of algebraic
212

surfaces in the following:
M is a minimal rational surface, then (c-.c-) - (8,4) or (9,3);
M is a ruled surface of base genus g, then (Cj,C2) = (8(l-g),4(l-g));
M is an Enriques surface, then (c.,C2) = (0,12);
M is hyperelliptic, then (Cj,C2) = (0,0),
M is a K-3, then (Cj,C2) = (0,24),
M is Abelian, then (c.,C2) = (0,0);
M is minimal elliptic with /c = 1, then (Cj,C2) = (0,n), n > 0;
is a minimal general type surface, then (c,,C2) = (ni,n), m,n > 0.
M
The following theorem says that to study the Chern number geography of
algebraic surfaces it is enough to look at surfaces in P .
Theorem. Every smooth algebraic surface M can be holomorphically embedded in
P^.
Proof. Without loss of generality assume that M lies in P° with n > 5. Pick a
point p € P°\M and consider the projection centered at p
IT : M -* P°"\ q € M H L ,
p ' ^ pq
where we think of P°~ as the space of lines in P° through p. The map ir will
be an embedding if there are no lines through p meeting M in at least 2 points
counted with multiplicity. Now the set of bisecants to M is parametrized by
(M X M)\{diagonal}.
It follows that the union of the bisecants to M lies in a subvariety of dimension
at most 5. It follows that we can choose p € M to insure that tt is a
*^ p
holomorphic embedding. The rest follows by repeatedly applying the projection
until n-1 becomes 5. d
213

§9. Complex Spaces and Singular Surfaces
By a ring we will always mean a commutative ring with a multiplicative
identity. A ringed space is a topological space X equipped with a sheaf of rings
Oy -* X, called the structure sheaf. A local-ringed space is a ringed space
(X,^ ) such that every stalk 0^ , p € X, is a local ring.
A A,P
Example. Let R be a ring. The spectrum of R is
Spec(R) = {jj c R: p, a prime ideal}.
Topologize Spec(R) by decreeing that the following sets are closed:
{V(I): I, any ideal in R}, where V(I) = {p € Spec(R): I C p}.
The structure sheaf, 0, on Spec(R) has as its stalks the localizations R , p 6
Spec(R), and is given by the following presheaf: For any open set U c Spec(R)
^(U) = {s: U -♦ II Rjj, s(p) € Rjj, s is locally a quotient of elements of R}.
(So given any p € U there is a neighborhood V of p and elements a,b € R such
that for every q € V, s(q) = ^ € R and b (^ q.) Then (Spec(R),<7) is a
local-ringed space.
Let B c C° be an open ball. A closed analytic subspace (Y,^^) o^ ^ ^^ ^'^y
local-ringed space which can be obtained as follows: Let {f.} be a set of
holomorphic functions on B, and
J = the subsheaf of 0„ generated as ^--module by the f.'s.
Now let Y be the common zero set of {f.} and 0^. — OUJ.
Definition. A complex space is a local-ringed space (X,^) for which X is
Hausdorff, second countable, and which is locally isomorphic to a closed analytic
subspace of an open ball in some C".
Let (X,^„), (Y,^y) be local-ringed spaces. Also let
f: X - Y
be a continuous map. The direct image sheaf (0^ -♦ Y is given by the presheaf
(f^<7^)(U) = 0^{r\v)), U an open set c Y.
214

A local-ringed morphiam is a pair of maps
f: X - Y, f: 0^ - f^O^
with the following properties:
1) f is continuous;
2) f is a sheaf homomorphism;
3) f : ^Y ff ^ ~* ^Y ^s * local-ring homomorphism.
The map f: ^ -♦ f^^ induces ring homomorphisms 0^{V) -* OyiC (U)) for any
open U. Taking the direct limit over U containing f(p) we get the maps
Mm yu) = ^Y,f(p) -* ^l^ 0^{r\v)) = 0^^^.
The condition 3) says that this map is a ring homomorphism taking the maximal
ideal of 0^ to the maximal ideal of f^^„.
Definition. Let (X,^^), (Y.^y) ^^ complex spaces. A local-ringed morphism
(f,f): X -* Y
is called a holomorphic map if f: 0^ -* (0^ is a C-algebra map.
I ♦A.
Let (X,^„) be any complex space. Let SH denote the sheaf given by the
presheaf {JH(U): U open in X}, where JH(U) is the ^^-module given by
{f € 0{V): f^ = 0 for some i € J"^}.
The complex space (X,<7^/iH) is called the reduction of (X,0^). If <7^/iH = 0^,
then we say that the complex space is reduced. Hereafter, we only consider
reduced complex spaces.
A reduced complex space X is irreducible if it can not be written as a
nontrivial union of closed subspaces. Any reduced space uniquely decomposes
into a locally finite union of irreducible subspaces, called the irreducible
components.
A point p € X of a reduced complex space is called a nonsingular or
smooth point if X is a complex manifold near p. The singular points of X form
a proper closed analytic subset of X, A complex space is a complex manifold if
215

and only if it is reduced and the singular locus is empty. (The structure sheaf of
a complex manifold is its sheaf of germs of holomorphic functions.)
Let p be a point of a reduced space X. Then the dimension of X at p is
simply the KruU dimension of the local ring 0^ , i.e., it is the supremum of the
lengths of all prime chains in ^ .
A,P
A point p e X of a reduced space X is called a normal point if the local
ring 0^ is integrally closed in its quotient ring. If every point of X is normal
A,p
then X is said to be normal. For a reduced normal space X we have
codim(X. ) > 2,
where X. denotes the set of singular points.
Let X, Y be irreducible complex two-dimensional spaces. A proper
holomorphic surjective map in X -♦ Y is called a bimeromorphic map if there are
analytic subsets A c X, B c Y such that ir. X\A -♦ Y\B is a biholomorphic map.
Given any normal complex two-dimensional space X there exists unique up to
biholomorphsm complex two-manifold M with a bimeromorphic map
?r: M -» X
such that M does not contain an exceptional curve of the first kind (i.e., a
smooth rational curve with self-intersection -1). The smooth surface M is called
the minimal resolution of singularities of X.
Definition. A curve (connected and compact) C on a smooth compact complex
surface M is called exceptional if there exists a bimeromorphic map
tt; M - X
with the following property: there exists p € X, a neighborhood tj of p, a
neighborhood U of C such that
t(C) = p; K maps U\C biholomorphically onto U\{p}.
We say that C contracts to the singular point p.
Theorem (Grauert). Let C be a curve on a compact complex surface and also let
216

(C.) be its irreducible components. Then C is exceptional if and only if the
intersection matrix (C.-C.) is negative definite.
Proposition. An irreducible curve C c M is an exceptional curve of the first kind
if and only if
C-C < 0, Kj^-C < 0.
Proof. Suppose C is an exceptional curve of the first kind. Now the adjunction
formula says
K^.C + CC = K^.
But C-C = -1 and K^ = -2. Conversely if C-C < 0 and K^^-C < 0, then
deg(K^) < 0.
Hence C is a smooth rational curve, and by adjunction C-C = -1. □
Hirzebmch-Jung Strings
An exceptional curve C c M is called a Hirzebruch-Jung string if it can be
written as a finite union of smooth rational curves C. such that
Cj-C. < -2, C.-C. = 1 if |i-j| = 1, Cj-C. ~ 0 if |i-j| > 2.
The simplest Hirzebruch-Jung string is a smooth rational curve with
self-intersection -2.
ADE Curves
These are the exceptional curves for which all irreducible components are
smooth rational curves with self-intersection -2. Let C = U C. be an ADE curve
with irreducible components {C.}. Since C.-C. = -2 and (C.-C.) is negative
definite we see that the intersection matrix (C.-C.) is given by the Cartan matrix
of one of the compact Lie groups A (n>l), D (n>4), Eg, E„, Eg. Observe that
the Cartan matrix of A represent a Hirzebruch-Jung string.
Proposition. Let C = U C. c M be an exceptional curve with K -C. = 0 for
every i. Then C is an ADE curve.
Proof. Since K -C. = 0 and C.-C. < 0 we have by the genus formula g(C.) <
217

0. So C. is smooth rational. By adjunction C.-C. = -2. □
Let X be a normal complex two-dimensional space, and M a smooth
complex surface, consider a double covering
r X - M
ramified over a curve B c M. The set ir~\B) c X is called the ramification
locus. It is not difficult to verify that X is nonsingular iff B is a smooth curve;
if B has a singular point given locally in B by
^(x,y) = 0 centered at p € B,
then X has a singular point at iT (p) given locally in X by the equation
w + f(x,y) = 0 centered at ir (p).
Definition. Let x; X -♦ M be as in the above. The singular point ir {p) is
called a nonessential singularity if p € B can be given locally by one of the
following equations centered at p:
(type A^, n>l) f(x,y) = x^ + y°+l = 0,
(type D^, n>4) f(x,y) == y{x^ + y^'^) = 0,
(type Eg) f(x,y) = x^ + y^ = 0,
(type E^) f(x,y) = x(x2 + y^) = 0,
(type Eg) f(x,y) = x^ + y^ = 0.
Theorem. Let X be a normal complex two-dimensional space with a nonessential
singular point p. Then we can find a smooth surface M, an ADE curve (of the
corresponding type) C C M, and a bimeromorphic map
such that C is the exceptional curve giving the singularity p. Conversely, we
suppose C c M is an ADE curve on a smooth surface giving a singularity p € X.
Then p is a nonessential singularity of the corresponding type.
For a proof see [BPV] pp. 87-90.
Nonessential singularities can be characterized in terms of the Chern
218

numbers of the minimal resolution of singularities as follows.
Theorem (Persson). Let t: X -♦ M be a double cover from a normal complex
two-dimensional space onto a smooth surface ramified over a curve B c M.
Then ? singular point q € ?r~ (B . ) is nonessential if and only if
c\{X) = (/(c^(Y) - I [B]))\ c^{X) = ^*(C2(Y) - I c^(B)),
where X is the minimal resolution of singularities of X.
See [P] for more on the above theorem.
219

Chapter V. Hermitian Differential Geometry
Techniques from Hermitian differential geometry are discussed in this
chapter; we give a moving frame theoretic sketch of submanifolds of projective
space.
§1, which is somewhat independent from the rest of the chapter, deals with
various metric and topological properties of the complex Grassmannian. A
topological significance of Grassmann manifolds is that they serve as classifying
spaces for vector bundles. Specifically, any complex vector bundle over a
manifold can be realized as the puUback by a continuous map of the universal
bundle on a Grassmannian. This correspondence along with an explicit
computation of the Chern class of the universal bundle is given in §1. In §2 we
consider holomorphic maps from a compact Riemann surface into projective
space, and derive so called the Pliicker formulae for space curves. The Pliicker
formulae for a space curve give relations amongst the genus and various extrinsic
invariants, namely associated degrees and ramification indices. In §3 we consider
complex submanifolds of projective space and examine various quantities arising
from the higher osculating maps of a projective complex submanifold. Weyl's
formula which expresses the first Chern form of an osculating bundle to
osculating Kahler forms is derived. Weyl's formula may be thought of as a
higher dimensional analog of the Pliicker formulae. The complex fundamental
forms of a complex immersion into projective space are also introduced in §3.
Gauss-Bonnet type formulae for projective hypersurfaces relating the total
curvatures to the Chern numbers are derived in §4. As an application we prove
that a smooth surface in P with an immersive Gauss map must be a quadric.
Finally in §5 we discuss a result of [T] characterizing the Veronese surface as the
only surface in P whose osculating sequence is (2,5).
220

§1. Grassmannians
Let G . = G(n,k) denote the Grassmann manifold of k-planes in C°. Also
PG . ^ G(n+l,k+l) denotes the Grassman manifold of projective k-planes in P°.
Note that P° ^ G(n+l,l).
Take A € G(n,k). We can write
A = [v^ A • • • A vj,
where the v.'s independently span A and [•] is the equivalence class coming from
>(v^ A • • • A Vj^) ~ (Vj A • • • A Vj^), X € C\{0}.
So A is represented by a decomposible k-vector in C°.
Choose a multiindex I = Oi»"*»^k} ^ {^t"',^}t and put
Uj = {A € G(n,k): A n A^_^ = 0},
where A _. is the (n-k)-plane spanned by {c: i t I}- In other words,
Uj= {A€G(n,k): c*^A ... A c*^(A) M},
where (c.) is the dual cobasis to the natural basis (c.) of C°. Any A in U. can
be written uniquely as a decomposible vector of the form
V, A • • • A V., V, = e. + J?c ,
1 k' 1 1 1 a'
where i € I, a € I*^ = {1,- • •,n}\I. U. is an open set in G(n,k) and
are called the standard coordinates. The matrix of (Vi.*".\) relative to {e.;e )
is given by
■(9"
{U-, {^.): |I| = k} makes G(n,k) into a complex manifold.
Schubert Cycles
Fix a flag V^ c • • • c V^ = C°, dim V. = i. For any A € G(n,k) we get a
nested sequence of subspaces
221

ocAnv, c---cAnv =A.
1 n
For a generic A we have
A n V. = 0 for i < n-k;
dim(A n V.) = i+k-n for i > n-k.
Take a sequence of integers a = (a ,• ••,*,) and set
W^ = {A € G(n,k): dim(A n V^_j^ .^ ) = i for every i}.
Since
dim(A + V ... ) = n-a.,
the set W =0 uidess a = (a .•••,a,) is a nonincreasing sequence of integers all
less than or equal to (n-k). We also have dim (A n V , . ) = i if and only
if the rank of the last (k+a.-i)>«k minor of a matrix representative for A is
exactly k-i. It follows that the closure
W. = {A. dim(A n V^^^.^p > i}
is an analytic subvariety of G(n,k). Also
W N (;k(n-k)-Eai
a -
and the sets
{W^: a^ > •.. > a^, a. < n-k}
give a cell decomposition of G(n,k).
Theorem. The integral homology of G(n,k) has no torsion and is freely generated
by the cycles
where a = {a.,- • -.a.} ranges over all nonincreasing sequences of integers between
0 and n-k.
Proof. There are cells in even dimension only; all boundary maps are zero. □
Notice that the cycles <t is constructed from a flag V = (V.) in C°. Any
two flags in C° are related by an automorphism € GL(n,C) of C°, hence the
homology class of <t depends only on the multiindex a not on the choice of the
222

flag, o — a (V) are called the Schubert cycles.
Example. The Schubert cycles on G(4,2) are
codimension 1: c. qCV.) = {A: dim(A n V.) > 1};
codimension 2: c^ ^(Vj) = {A: A c Vj}, c^ ^^ = {A: A 3 V J;
codimension 3: a^ ^^ C V^) = {A: V^ c A c Vj}.
Recall that the universal bundle S -♦ G(n,k) is the holomorphic subbundle
of C°*G(n,k) given by
S. = the subspace A c C°.
The exact sequence
0 - S -» C°«G(n,k) - Q -» 0
defines the universal quotient bundle Q -♦ G(n,k).
The following theorem can be proved via direct integration. For a proof
see [GH] pp. 410-411.
Theorem. Let \a\ denote the homology class of <t , where a = (1,1,«'«,1) (i
times). Also let \a ]* denote the Poincar6 dual of [<t ]. Then
CjCS) = (-i)Vj*.
Let M be a compact smooth manifold of real dimension < 2n-2k. It can be
shown that (cf. [BT] p. 299) the isomorphism classes of complex vector bundles
over M of rank k are in bijective correspondence with the homotopy classes of
continuous maps f: M -♦ G(n,k) via
f H r^S.
Using this result we can compute the Chern classes of a complex vector bundle of
rank k, E -♦ M, as follows: Let f: M -♦ G(n,k) (n sufficiently large) be a
continuous map with
r^s = E.
Then
c.(E) = c.(r^S) = f*oc.(S).
223

Another application is the
Integrality Theorem. Let L -* M be a complex line bundle. Then c (L) is an
integral homology class yrithout torsion.
Proof. Take a continuous map
f: M -» G(n,l) = IP""^
with r^S = L. Now S = H*, where H -♦ IP°~^ is the hyperplane bundle. We
know that c.(H) is a generator for H (P ,2?) ^ 2?. The result follows since
c^(L) =-f*(c^(H)). D
Let M be a compact complex manifold, and suppose we are given a
holomorphic embedding
f: M -» IP°.
Then the line bundle F H -♦ M is a positive line bundle on M since H is positive
and f is an embedding. The Kodaira Embedding theorem says that an arbitrary
compact complex manifold M can be holomorphically embedded into a projective
space if and only if there is a positive line bundle on M. In fact, given a
positive line bundle L -* M the associated rational map (p, .■, n sufficiently
large, gives a projective embedding.
Let A = [v. A • • • A V. ] 6 G(n,k). Letting (c.) denote the usual basis of
C° we can write
V, A ••• A V. = S P e A ••• A c (P, antisymmetric in (a.)).
Q
The Pliicker embedding
p: G(n,k) - pN, N = Q - 1,
is given by
A H [P 1.
Theorem A. The KShler form of (G(n,k),ds ) is equal to ir times the Chern form
of the pullback line bundle by p of the hyperplane bundle.
224

p*H —♦ H
1
G(n,k) -^ pN
2
The metric ds will be defined later in this section and the proof of
Theorem A will be given shortly thereafter.
Unitary Frames
In C° we have the standard Hermitian inner product
<x,y> = x^y^ + ... + xj^.
Analogously we can define a Hermitian inner product in A C , the set of
k-vectors in C°. (Of course, a'^C" ^ C^"^\ N+1 = Q.) For decomposible
k-vectors
A, = V, A • • • A V., A„ = w, A • • • Aw.
11 k' 2 1 k
define
<A ,A,> = det <v.,w.>.
It is easily checked that this definition is independent of the particular vectors
v.,w. chosen to write A and A . Extending the above definition linearly to all of
A C" one obtains a Hermitian inner product on it. We put
|A| = +<A,A>^/2.
A unitary k-frame in C° is simply a collection of vectors (e,,-",e.) in C°
such that
<e.,e.> = 6...
Via the usual basis (c.,- • •,€ ) we identify
{unitary n-frames in C°} ^ U(n).
There are the fibrations
225

JTj: U(n) - St(n,k),
r^. St(n,k) - G(n,k),
r = ir^oTTy U(n) -♦ G(n,k)
(St(n,k) denotes the Stiefel manifold of all unitary k-frames in C°), where
Ti(ei,---,e^) = (ej,.--,e^),
TjCe^.-'-.e^) = [ej A .-. A ej.
The projection ir makes G(n,k) into a homogeneous space U(n)/U(k)>«U(n-k).
Let n denote the u(n)-valued Maurer-Cartan form of U(n). Using the
matrix coordinates (x°) of U(n) and keeping in mind the usual tangent space
identification for matrix groups we can write
Rewriting
n = (np, n;j = (x-^);dx;.
dx = x« • nf
a p a
Exterior differentiation of both sides of the above equation leads to the
Maurer-Cartan structure equations
dn = -n A n.
Put u = e*n, where e is a local section of U(n) -♦ G(n,k), We then have
de = e- • or.
a p a
Put A = A/|A|, where A is a decomposible k-vector in C° and write
\ = e^^ ... Ae^
for a unitary k-frame (e.,...,e.). We compute that
<dA^,A„> = S 4
where u denotes e*n and the local section e is chosen such that the first
k-vectors of e are (e.,. • .,e.). Upon exterior differentiation, we find that
<dAp,dAp> = (S a/).(E a^) + S oft-uft,
where 1 < i < k, k+1 < a < n. It follows that
<dA„,dA^> - <dA„,A„><A„,dA^> = S iol-U^I,
226

or equivalently,
(<A,A><dA,dA> - <dA,A><A,dA>)/|A|^ = E w'-w' = ds^
The left hand side of the above equation shows that ds is Hermitian and the
right hand side shows that it defines a positive definite metric on G(n,k). The
expression is independent of the choice e to write w. To put it another way, the
symmetric product ds is Ad(U(k)>«U(n-k))-invariant.
Put 0 = [cjA- • -AcJ € G(n,k). U(n) acts on G(n,k) via
g-[VjA---AvJ = [g(VjA---Av^)],
and the isotropy subgroup of this action at 0 € G(n,k) is
^0 = ^(o;b)= ^ ^ U(^^)' ^ ^ U("-^^)> = U(k)xU(n-k).
Let ^ denote the Lie algebra of G., and also let m denote the orthogonal
vector space complement with respect to the Killing form. So
g = ^ e m.
Note that m may be identified with the tangent space T G(n,k) via ir .
The negative of the Killing form restricted to m, upon translation, defines a
U(n)-invariant Hermitian metric on G(n,k). Moreover, any invariant Hermitian
metric on G(n,k) is a constant multiple of this metric.
Since G(n,k) and P are both Hermitian symmetric, and since on a
Hermitian symmetric space any two invariant metrics are constant multiples of
2 * 2 2
each other, ds must be a constant multiple of p dSpg, where dSpg denotes the
Fubini-Study metric on P .
The Kahler form of (G(n,k),ds ) is given by
$ = i E w? A w? = ir £ w? A w' = ;Jr d(S a/).
2 1 1 2i 1 a 2i ^ r
2
From this we see directly that ds is KShler.
Since
E oi = (5- 5) log|A|
we also have
227

$ = i55log|A|.
Local Hermitian Geometry
Let E -♦ N be a Hermitian holomorphic vector bundle of rank n. We saw
that there exists a unique type (1,0) metric connection (sometimes called the
Chern connection)
V: r(E) - r(T*M • E).
The connection matrix, 6, is defined relative to a local frame (e ):
Ve = e. • /.
a p a
The curvature matrix, x. is defined by
X= dO + e h e.
Suppose e = e-g, (g € GL(n,C)-valued) is another local frame, then
0 = g"Mg + g'^eg, X = g"^ X g-
Put h . = <e^,e„> = h° The tilded quantity h.„ is defined similarly
using the frame e. We then have
h = *ghg.
Also
dh = *wh + hw, 0 = *nh + hjl.
When e is unitary, i.e., h = ^ we obtain
*w + a; = 0, *n + fl = 0.
Take a local holomorphic frame s = (s.,'«',s ) of E -♦ M. Recall that
>
relative to s
w = H"^5H, n = -H"^5H A H"^5H + h"^55H.
For a line bundle, the second of these equations simplifies to
Q = 88 logh.
228

Let $ denote the KShler form of (G . ,ds ). Given a line bundle endowed
with a connection we will call i/2ir times its curvature form, the Chern form.
Proof of Theorem A. The line bundle
a'^C" - pN, N+1 = (J),
is just the universal bundle S -♦ P . On S there is the Hermitian inner product
introduced earlier, namely, the one coming from the norm |A|. The norm |A|~
induces a Hermitian structure on S~ = H, the hyperplane bundle. The
curvature form of
p~ H -♦ G(n,k), p = the Pliicker embedding,
is given by
X= dd logh,
where h = |A| . On the other hand,
$ = i55log|A| = -i55log|A|,
and the result follows, d
Let e = (e,,"",e ) be a local section of U(n) -♦ G(n,k), and put, as usual,
uj = e*n. We then have
$ = I S w' A 0? = the Kahler form of G(n,k),
X = £ w? A w? = the curvature form of p~ H,
where on H we use the metric coming from the norm |A|.
Consider the projection
ir. U(n) - St(n,k).
ir(e) = (6,,'",e.) gives a local frame for the universal bundle S -♦ G(n,k). The
universaal bundle S -♦ G(n,k) is naturally a Hermitian bundle and T(e) is a local
unitary frame for it. Now
de. = e. 9 (jjI,
where 1 < i,j < k. From this we see that (w!) are the connection forms of (S,Vg)
229

with respect to T(e).
The curvature forms are given by
Y. = do/. + oA A w. = - S w Aw*, k+1 < a < n.
The matrix x = (X-) is skew-Hermitian and we find that det(I + six) is real.
Consequently,
c.(S) = [Re V\^x)]-
Direct integration shows that
1
Re A^X) = 1.
where a = (1,«-«,1) (i times). In other words, [Re ?(•)] is the Poincar^ dual of
the Schubert class [a ] times (-1):
(-I)VJ* = c.(S) = [Ite V\J^)] € H2'(G(n,k)).
The upshot is that the Chern classes c.(S) can be computed in terms of the
Maurer-Cartan form using only local computations.
The Curvature Forms of (G(n,k),d82)
We explicitly identify G(n,k) with U(n)/U(k)xU(n,k) via G^ =
U(k)>«U(n-k). The following index ranges will be used:
1 < h,i,j < k; k+1 < a,b,c < n; 1 < a,/3,7 < n.
On G(n,k) we use the metric
ds = E wf • w*
where w = e*n = e~ de, and e is a local unitary frame. The forms (w?) are all
of type (1,0) forming a local unitary coframe on G(n,k). We vrill write the
connection and the curvature forms relative to it.
Put w" = wf. We then have
bj '
4 = <i + Kh * Ci-
230

where 6 is the connection matrix and x is the curvature matrix. From the
Maurer-Cartan structure equations of U(n) we can compute 6 and x *s follows:
dw" = dojt = - if: K UK - uj^ h uj.
1 J 1 b 1
= -(if - (}^ h wt - iJt K u). - S w' A w^
So
and
dw" = -(w* - a/) A w" - Sw! A w'J - S (jj A w^\
bj J b b 1
To compute the curvature forms
d^! = i{-Jt hut-iJ^Kul)- tA-J h Jt - J hif:).
bj J h b c b' b^ h i c r
On the other hand
ch bj ,^ h c c r ^ 1 b b h'
c,h
So
= S ((f wi A J? + ^. w' A U?).
c,h ^ ^ J J *= ^
Xf! = S <f 0/^ A w^ - ^ w' A of.
'^bj , b c i J h b
h,c
Theorem. The first Chern form of the holomorphic tangent bundle over G(n,k) is
equal to (N+l)/ir times the Kahler form of G(n,k), where N+1 = (f).
Proof. We have
trace(x) = S Xj
i,a
= S ( S w' A w? - S wf A a/')
. ^ c 1 , h a'
i,a c h
= (N+1)(S u/t A uft).
Now the first Chern form of T ' G(n,k) -♦ G(n,k) is given by
^•trace(x)
and the KShler form of (G(n,k), ds ) is
^ S w? A of. □
2 1 1
231

We can give another proof of the above result using Theorem A as follows.
Put
$ = the Kahler form of (G(n,k), ds^),
,1,0.
rj(TG^ k) ^ *^® ^"* ^^^^^ ^^^^ °^ T''"G(n,k),
Theorem A then gives
T^{p ^H) = the Chern form of p ^E -* G(n,k),
We know that
$ = ir-T^{p~^E).
Cj(TlpN) = (N+l)c^(H).
Pulling back the Chern forms to G(n,k) via p we get
^l(T^n,k) = (N+l)ri(/;-'H).
Therefore,
^l('r^n,k) = ((N+1)/t)$.
§2. Space Corves: the Plucker Formulae
In this section we consider holomorphic maps from a (compact) Riemann
surface into P". Any nonconstant holomorphic map from a Riemann surface M
into P" induces on M a Hermitian metric with "mild" singularities, and it is not
difficult to generalize the one-dimensional Hermitian geometry to accomodate
such metrics.
Definition. Let U be a domain in a Riemann surface M. A smooth function
f: U - C
is said to be of analytic type if for each x € U, if z is a local holomorphic
coordinate centered at x, then
h = z^h.
»+ t
where b € J , h is a smooth function with h(0) ^ 0.
232

It is not difGcult to show that the functions of analytic type are exactly
solutions of exterior equation
dh = }xil> (mod dz),
where V is a C-valued 1-form on U. So if h is of analytic type on U, then h is
either identically zero or its zeros are isolated and of finite multiplicity b.
Definition. A type (1,0) form V on U c M is said to be of analytic type if in a
neighborhood of every point of U, V can be written as the product of an analytic
type function and dz, where z is a local holomorphic coordinate. A positive
semidefinite Hermitian inner product in the holomorphic tangent bundle TM is
called a singular metric if it can be given locally as
ds = V • i^,
where V is a, not identically zero, type (1,0) analytic form. The singular divisor
of (M,ds ), denoted by D , is defined to be the zero divisor of V, i-C,
D = S ord (V)p.
^ p€M P
The divisor D„ is locally finite and is an honest (meaning finite) divisor if
M is compact. The degree of D- is the total number of zeros of V on M counted
according to multiplicity. Note that the zeros of iff are well-defined on all of M,
albeit V is defined only locally.
Let M be a Riemann surface endowed yrith a singular metric
V* • ^ = h(z) dz • dz, h(z) > 0.
We have the following equations away from the support of Dg:
dif=-ehi), X = d^' = IK ^ A ^,
where 6 is the complex connection form, x is the curvature form, and K is the
Gaussian curvature.
From [Y] (Chapter 5, Section 2) we have the
Gauss-Bonnet-Chem Formula for Singular Metrics. Let M be a compact
233

Riemann surface of genus g endowed with a singular metric. Then
2?l„X=2-28 + 'ieg(Ds).
Let M be a Riemann surface and consider a holomorphic map
f: M - IP°.
To avoid redundant considerations we assume that f is nondegenerate, that is to
say, f(M) does not lie in a lower dimensional projective subspace. In a
neighborhood of any point, f can be holomorphically lifted to C""^ \{0}. Let
v(z) = '(v%),---,v°(z))
be such a lifting, where z is a local holomorphic coordinate on U c M. So the
v''s are holomorphic and [v(z)] = f(z). Put
^0 = V, e^ = V' = g.
Suppose we have another lifting v of f, v = (v ,*'*,v°). Then we must have,
since [v] = f, v' = >v' for every i for some C*-valued function \. Put
gp = V, Cj = V'.
Now Cj = \x'y' + Xy') = \x'y') + Xe^ Thus
Cq A gj = >2 e^j A ej,
and the two-plane [e^Ae ] (assuming that e.Ae. ^ O) is well-defined independent
of the choice of a homogeneous lifting. Put
fj = [e^AeJ: M\E -* G(n+1,2) c P^, N = f+l) - 1,
where S denotes the isolated globally defined zero set of e.Ae., and G(n+1,2) is
included in P via the Pliicker embedding. Since f. is a rational map on a
Riemann surface it extends uniquely to a holomorphic map defined on all of M.
Definition, f.: M -♦ G . g = ^°* (^ ,, 2» *^® Grassmann manifold of complex
two-planes in C""^ , is identified with the space of projective lines in P°) is called
the dual curve of f or the first associated curve of f.
Maintaining the above notation let
234

There are also the tilded quantities:
g^ = \x'y' + >v''),
§2 = \x"y' + 2>'v'' + Xy").
We see that
[Cq a gj A ej = [e^ A ej A ej
whenever e.Ae.Ae, ^ 0 so that [ ] makes sense. The nondegeneracy assumption
on f again guarantees that the zeros of e.Ae.Ae are isolated, hence they can be
removed. The second associated curve of f is the holomorphic map
fj = [eoAejAe^]: M -* G(n+1,3) c P^, N = (°+^) - 1.
Recursively proceeding we obtain
^k = IV-'^^l- ^ -* G(n+1M1) C P^, N = {I'll) - 1.
where 0 < k < n-1, f = f = [e ].
Recall the explicit homogeneous space identification
IP° = U(n+l)/Go, Gq % U(l).U(n)
coming from t. U(n+1) -♦ P°, i^i^Q,"' fij) = [Sq]-
Definition. A local section e = (e , •••, e ) of f~ U(n+1) -♦ M is called a
Frenet fcame along f if
[e.A- • -AeJ = fj^ for every k.
Fix a Riemannian metric, ds^., on M in its conformal class. Locally we
write
dSjj = <P'<p
for some nonvanishing type (1,0) form <p. We have from [Y] Chapter 5
Proposition. Let e be a Frenet frame along a nondegenerate holomorphic curve
f: M - P°,
and put u = e*n, where fl is the u(n+l)-valued Maurer-Cartan form of U(n+1).
235

Then
^l'= ^l'= ••• =^-2'=0' >. >i+l;
where Z]_., 1 < i < n, are all local C-valued analytic type functions. Moreover,
the real valued functions (r*) = Z!_ Z!_. are globally defined not identically zero
analytic type functions on M.
Exterior differentiation of the above equations leads to so called the metric
structure equations for a projective curve: Let f: M -♦ P° be a nondegenerate
curve. Then away from the zeros of the r 's we have
A log r' = K + 2(r'-V - 4(1')^ + 2(r'+V,
where 1 < i < n, r = r = 0, K is the Gaussian curvature of (M,ds ), and A
is the Laplace-Beltrami operator of (M,ds ).
For the rest of this section we assume that M is a compact Riemann
surface of genus g. Consider a nondegenerate holomorphic map f: M -♦ P°. We
know that each f. (M) is an algebraic curve by a variant of Chow's theorem. For
0 < k < n-1 put
d^ = deg f^(M) c pN, N = (J+J) - 1.
Let p € M and also let #(p) denote the ramification index of f at p.
Using the inhomogeneous coordinates write
i{z) = \l, f\z), ...,f°(z)].
Then #(p) = min^ ord (f°). Put
#0 = #= S #(p).
p€M
#. is the total ramification index of f. Similarly define #. (p) and #. for f..
The reader may verify that #. is precisely the number of zeros of r counted
with multiplicity.
Definition. The k-th osculating metric of f is a singular metric on M given by
236

where u = e*n, and e is a Frenet frame. Its KShler form is
2 N 4
Let ds-. denote the normalized Fubini-Study metric on IP . A standard
calculation shows that
* 2 2
fu ds„ = ds..
k N k
Consequently
Aj^ = f^(the Kahler form of (P^, ds^)).
Put <p = Jy^ and also put
D. = the singular divisor of ds.
so that
deg D, = #^,
We now compute the connection form and the curvature form of (M, ds.)
relative to <p . Using the Maurer-Cartan structure equations of U(n+1) we
obtain
d^ = da;j;+^ = (a;j; - c^+J) A /.
So 6, = 0^^: - 0^ is the complex connection form of (M,ds.). Exterior
differentiation of both sides of the above equation leads to
,a k-1 . -k-1 , rt k . -k k+1 . -k+1
= 2i(A^_, - 2A^ + A^^j).
Recall also that
k ~W
where K. is the Gaussian curvature of (M, ip -ip). We have the
Plucker Formtilae. Let f: M -♦ P° be a nondegenerate holomorphic map from a
compact Riemann surface M of genus g. Then
2g - 2 - #, = d,_j - 2d, + d,^j,
where 0 < k < n-1, d_j = d^ = 0, d^ = deg f(M).
237

Proof. We have
1
M
A. = the area of (M, ds.).
M *
The Wirtinger theorem states that
The result now follows easily, o
Note that we could have integrated the metric structure equations directly
to obtain the Pliicker formulae.
Remark. Let M be a Riemann surface. Given a holomorphic map
f: M - IP° = U(n+l)/U(l)xU(n) = SU(n+l)/S(U(l)xU(n))
the totality of its Frenet frames defines a global holomorphic map
^^ M - SU(n+l)/T,
where T = S(U(1)°"^^) is a maximal torus in the simple Lie group SU(n+l).
There is a natural decomposition of the tangent bundle of SU(n+l)/T:
T(SU(n+l)/T) = ^ ® V,
where 7 -* SU(n+l)/T, called the (super-) horizontal distribution, is a rank n
holomorphic vector bundle generated by the simple root spaces of SU(n+l). And
it is not hard to show that $, is horizontal, i.e., tangential to the distribution 7.
Conversely, a horizontal holomorphic curve in SU(n+l)/T projects down to give
a holomorphic curve in P°. Thus projective curves may be thought of as
horizontal curves in the unitary flag manifold SU(n+l)/T. Indeed this point of
view can be generalized to study horizontal curves in G/T, where G is any
compact simple Lie group and T a maximal torus in G. The interested reader
may consult [Y3] for further materials on horizontal curves in G-flag manifolds.
238

§3. Complex Submanifolds: Weyl's Formulae
Let M be an m-dimensional complex manifold and consider a holomorphic
map
f: M - IP°.
There is a local holomorphic lifting
f: U c M - C°+^{0}
z = (zVf(z) = *(!%),...,f»),
where z = (z) are local holomorphic coordinates.
r.n+1
c°-^'\{o}
/ i
U c M -» IP°
The first order osculating space of f at p € M is defined to be the subspace
of C°+^ given by
t(^) = space {f, 5f/djz'} .
At a generic point the dimension of T^^ is equal to m+1. For the rest of this
section we assume that the dimension of T^ ' is everywhere m+1. Note that
dim T^ ^ = m+1 if and only if f is an immersion.
We thus obtain a holomorphic vector bundle of rank m+1
t(i) -* M.
We also define a line bundle T^ ^ -♦ M by
t(°) = span {f}^ c e^\
i.e.,
t(°) = rh = r^H*,
where S -♦ P° is the universal bundle.
239

Put e. = f/|f| and choose local vector fields e,,"«,e so that
0 ' ' ' 1' ' m
(e.,ep'",e ) form a unitary frame of T^^ -♦ M. Define a rank m bundle, A^^
-♦ M, by
A^ ' = span {e.,*- •,e }.
We thus have an orthogonal decomposition
t(i) = t(°) e a(i).
The first osculating map of f is defined to be
f^^): M - G(n+l,m+l), p h T^^^.
t(i) -. s^
i i
M -4 G(n+l,m+l)
Note that
t(i) = f(iHs^,
where S -♦ G(n+l,m+l) denotes the universal bundle.
The second order osculating space of f at p € M is defined to be
t(2) = span {f, 8{/8z\ fi^f/te'teJ} .
A simple counting shows that
dim t(2) < m+1 + C^f).
Near a generic point of M the dimensions of osculating spaces are constant,
and we are primarily interested in the local picture at the moment. We assume
that the the dimension of T^ ' is constant throughout M. Put
dim t(^) = tg+l.
Choose local vector fields e ■,.**',e. so that (Cq."'.©! ) forin a local unitary
frame for T^ ^ -♦ M. Define a rank t -m bundle, A^ ^ -♦ M, by
A(2) = span{e^^j,...,e^^}
240

so that
t(2) = t(^) e a(2).
Recursively proceeding we obtain
tW = span {f, 5f/5z\ ..-, di^/dz'^.'-dz'"},
dim tW = t +1,
where we assume that each t is constant. We also define integers s by
dim A^') = t -t , = s .
a a—1 a
For notational symmetry we put
*0 = ^' ^0 ^ ^' *l = in, Sj = m.
The osculating order of /, denoted by o(f), is defined to be the smallest
integer such that
*o(f) = *o(f)+l' ^^^ *o(f) * *o(f)-l-
We assume that
T(o(f)) = c""^^
This amounts to assuming that f(M) is nondegenerate.
We call the strictly increasing monotone sequence
the osculating sequence of f. Observe that we always have
For example, for surfaces in P the two possible osculating sequences are (2,4)
and (2,3,4).
For each index a we have the following commutative diagram:
t(') — s
i i
M —. G(n+l,t^+l)
241

The bundle S -♦ G(n+l,t +1) is the universal bundle, and the bundles T^'^, S
are related by
t(*) = f(*)~^s.
a'
Definition. Let f: M*" -♦ P" be a nondegenerate holomorphic immersion with
constant dimensional osculating spaces. A local section e = (e.,««',e ) of the
bundle f" U(n+1) -♦ M is called a Frenet frame along f if for every a,
[e.A---Ae, 1 = t('), 0 < a < o(f).
By decreeing the Icoal sections to be Frenet frames along f we obtain a
U(sJ*U(sJ«' • •«U(s /jN)-principal subbundle of F U(n+1),
7- M,
which we call the Frenet bundle.
Let n = (n°) denote the skew-Hermitian Maurer-Cartan form of U(n+1).
We want to describe e*n, where e is a Frenet frame along f. Put w = e*n.
Then the forms u = (w°) satisfy
de = (/ • e^, 0 < a,/3 < n.
Now de^j = d(f/|f|) and from the definition of T^^' we see that de^ is T^^^
-valued. It follows that
£^ = 0 for /3 > tj.
In general, d(e., • • • ,e ) are T^*"^ ^-valued, and consequently
(*) (/ = 0, when a < t and ^ > t ,,.
^ ' a ' - a "^ a+1
As we shall see later in the chapter the above equations, upon exterior
differentiation, lead to the complex second fundamental forms of the immersion.
The matrix u = (oj°) looks like
p
242

x^
0
0
*
x^
x^
0
0
*
x2
• • •
x^i
x'
*
x'
(I = 0(f)),
where X* = (w,*) and X* are skew-Hermitian,
t ,+1 < I ,J < t .
a—1 a' a ~ a
For the rest of this section we will use the foUwing index ranges, where the
index a is fixed:
t ,+1 < I,J,K < t ;
a—1 - » » - a'
t,+l < i',j',k' < t,^j;
'.+.+1 <- ■'.J'.k' <- '.+2-
Recall that if s is a local section of U(n+1) -♦ G(n+l,t +1), then the
forms
*«-.a\
(s nr), 0 < i < t , t +1 < a < n,
give a type (1,0) unitary coframe on G(n+l,t +1).
Pull back the forms (0?) to M via a Frenet frame e, and write w = e fl.
By virtue of (*) the only nonzero forms among (w?) are
Let (e°!) denote the connection forms of (G(n+l,t +1), (s*n?) = (s O")).
We compute that
Pulling back the forms (^0^ - ^ft"?) to M by the Frenet frame e, and consulting
(*) again, we obtain the following nonzero forms:
243

We let the forms (^i,) define a connection on the rank s -s ,, bundle
^ jy a a+l
by letting them act as the connection forms relative to the unitary frame
(< • ej,).
This connection will be denoted by V^*'.
The curvature forms of (A^*)* • A^*"^^), V^*)) are given by
We want to compute the forms x ^^ terms of e*n = lj. Have
d^jlj = d(^j.ji - ^jlu^)
and
Thus
(t) Xjl] = ^jIC^o A J^' + o;^. A 0/^') - Sy^' A c^, + u^ll A c^l).
Let rj(v(*)) denote the first Chern form of A^*)* • A^*"^^) relative to the
connection V^*'. Also let $ denote the KS.hler form of the a-th induced metric,
a '
ds , which is, by definition, the puUback by the a-th osculating map f^*' of the
usual metric on G(n+l,t +1).
2 .
f(')
Remark, ds is, strictly speaking, not a metric on all of M since f^ '' is in
general not an immersion. However, f^*' is holomorphic and the branch locus of
f^*' is a proper analytic subvariety of M.
The following formulae were first written down in [T].
244

The Generalized Weyl Formulae. Let f: M*" -♦ P" be a holomorphic immersion,
where we assume that the dimensions of the osculating spaces are all constant
and that the highest order osculating space is C""^ . Then
Proof. By definition
r,(»W) = ij trace(x) = ij S xjij.
Also
$ = i S ti// A u//.
a 2 I I
(Note that the forms (a^ ) are not necessarily independent.) The result now
follows from (f). □
We also have
Theorem. Let f: M -» P" be as in the above theorem. Then
,_(tW) = -i *,.
where r^(T(*') denotes the first Chern form of T^*) -♦ M.
We will prove the above theorem in the following slightly more general
context.
Theorem. Let F: M -♦ G(n,k) be any nonconstant holomorphic map. Also let S
-♦ G(n,k) be the universal bundle. Then
where r denotes the first Chern form and $ denotes the Kahler form of the
induced metric on M.
f-^U(n)
i
U c M
—»
—»
U(n)
i
G(n,k)
Proof. Let fl = (0°), 1 < a,P < n, denote the skew-Hermitian Maurer-Cartan
245

form of U(n). Take a local section e = (e.,*• -.e ) of f" U(n) -♦ M and put u =
e n. Then
2.1 i'
a,i
where we use the index ranges: 1 < h,i,j < k; k+1 < a < n. Now (e.) (the first k
vectors of e) form a local unitary frame for the bundle f S -* M, and
de. = (J? 9 e = iJ. 9 e. + uj^ 9 e .
1 1 a 1 J 1 a
We thus see that {u^) are the connection forms of (F S,(e.)). The correspoding
curvature forms are
X^. = dJ. + J h J^ = - uP h Jt.
'^i 1 h 1 a 1
Now
r,{r'S) = L trace(x) = ^ So.^ A Jt = "1.$. u
a,i
Consider the holomorphic line bundle
dH = H*^ - P'.
We saw that there is an identification
H°(P',<7(dH)) = S'^(C'+^*),
where S (C'"^ *) = S ''"^ denotes the space of homogeneous polynomials of
degree d in r+1 variables. A polynomial F € S '"^"^ defines a degree d
hypersurface in P', and every hypersurface of degree d in P' arises in this way;
two polynomials F,G € S '"^"^ give the same hypersurface if and only if F = >G
for some X € C*. It follows that the linear system of all degree d hypersurfaces
(effective divisors of degree d, to be more precise) in P' is identified with
P(H (P'^,^(dH))), or what is the same, the complete linear system |dH|. A basis
of S'^''""*'^ is given by
{xiV'-.-x'^-i. > 0, Si. = d}.
I 0 1 r J - ' J '
Thus
246

dim ?{E\?',0{dE))) = ('^"J"') - 1.
The Linear System of Complex Second Fundamental Forms
We consider a holomorphic immersion
f: M*" - P".
Let e: U c M -♦ U(n+1) be a Frenet frame along f, and put
(J = e*n,
where n = (Q°), 0 < Q,0 < n, denotes the Maurer-Cartan form of U(n+1). This
means that
(*) '^ = 0, when o < t^, /? > t^^^,
where (t ) is the osculating sequence of f. In particular, we have (f is immersive
iff tj = m)
wj = 0, m+1 < a < n.
Exterior differentiation of the above equations, using the Maurer-Cartan structure
equations yields
w' A wj = 0, 1 < i < m.
Cartan's lemma implies that
wf = Qt.uA
I ^ij 0
for some local complex-valued functions Q*. = Q* on M. Thinking of (w') as
the homogeneous coordinates on P*"" the equation
Qf.t^wi = 0
^ij 0 0
defines a quadric hypersurface in P*" at each p € U c M. This quadric will be
denoted by Q'(f).
The n-m symmetric products
uft'Ljl = Qt.tA-Ljl,
i 0 ^ij 0 0'
are called the complex second fundamental forms of the immersion f. They are
smooth local sections of the second symmetric power of the holomorphic
247

cotangent bundle of M.
The usual local differential geometric method for extracting information
about the submanifold f is to normalize the complex second fundamental forms
thereby obtaining geometric invariants. On the other hand it may be interesting
to take a different perspective and consider the totality of second fundamental
forms.
Given a holomorphic immersion f: M*" -♦ P" we define the linear system of
second fundamantal forms at p, denoted by |II |, to be the linear system of
quadrics in P generated by {Q*, m+1 < a < n}.
Example. Recall the Veronese embedding
f = ^|jjj|: P*" - P^, N+1 = dim HV'",<7(dH)) = (^+'").
We leave it to the reader to verify that at any point p € P"* the linear system
III I is the linear system of all quadrics in P*" .
An interesting study of the linear system of second fundamental forms is
presented in [GH2].
§4. Projective Hypersurfaces and Their Chem Nnmbers
Recall that two submanifolds S., S„ C M are said to intersect transversally
if
T S, + T S„ = T M for every p € S, n S..
pi p2 p •''^Iz
When S. and S- intersect transversally we have
codim 8.082 = codim 8, + codim 82-
As a consequence
N = N ® N
where Ng denotes the normal bundle of 8 in M, i.e.,
248

Ng = TM|g/TS.
Suppose M is oriented and further suppose that S. and S. are closed oriented
submanifolds so that their Poincar6 duals make sense. Then
where (p„ € H"~"(M) (n-s = codim S) denotes the Poincar6 dual.
Let E -♦ M be any smooth vector bundle. Put
S. = the image of the zero section c E.
A section s: M -♦ E is said to be transversal if its image s(M) = S intersects S.
transversally. We have
Lemma. Let Z c M denote the zero locus of a tranversal section s of E -* M.
Then Z is a submanifold and
For a proof see [BT] p. 134.
In the above lemma if E and M are oriented, then Z is naturally oriented
so that
E|2®TZ = TMI2
has the direct sum orientation.
Example
Let M be a degree d smooth hypersurface in P". Consider the holomorphic
line bundle
H*^ - P",
where H = S* is the hyperplane bundle. Then M can be realized as the zero
locus M = Z^ c P" of a transversal section s € H°(P°,<7(H*^)). In other words, M
is a smooth divisor in the complete linear system |dH|. Recall that the linear
system of all degree d (possibly singular) hypersurfaces is parametrized by
P(hVAH*'^))) = |dH|.
Moreover,
249

H**^L, ® TM = TIP"L,.
We now compute the Chern classes of a smooth hypersurface M c P" of
degree d. From the above direct sum and the Whitney product formula we
obtain
c(TP")|^ = c(TM).c(H*^)^.
Now c(H®'^) = 1 + d(Cj(H)), and so
(*) (1 +h)"+l = (1 + Cj + ... +c^J(H-dh),
where h = Cj(H)|j^, and c. = c.(TM). Note that h" = 0.
Lemma. h""^([M]) = d.
Proof. The Poincar6 dual of a P c P" is given by
where $ is the Kahler form of the Fubini-Study metric normalized so that
Vol(P") = tt".
We also saw earlier that
Cj(H) = [i$] € h2(P").
It follows that h"~^ is the puUback of the Poincare dual of a P^ c P". Now
h"-'((Ml) = I h-' = j li,..*"-'l„l
= #(M,P ) (# = the intersection number)
= deg(M) = d. □
For M^ c P' we obtain from (*)
1 + 4h + Gh^ = 1 + (Cj + dh) + (dhc^ + Cj).
9 9 9 9 9
Write, as usual, h = h ([M]), c. = c.([M]), and so on. Since h = d we then
obtain
(la) cj = d(4 - df,
(lb) Cg = d(d^ - 4d + 6).
We now look at m' c P^. From (*)
250

5h = Cj + dh,
lOh^ = Cj + dhCj,
lOh' = C3 + dhCg.
Using the fact that h = d we obtain
(2a) cj = (5 - d)3d,
(2b) CjCg = d(5 - d)(d^ - 5d + 10),
(2c) C3 = d(-<i' + 5d^ - lOd + 10).
Remiark. The Chern numbers of a smooth complete intersection surface can be
computed in a similar fashion. See, for example, [P2] p. 205.
In the remainder of this section we will give a curvature theoretic
description of the Chern classes of hypersurfaces in P°. With possible future
applications in value distribution theory in mind we consider holomorphic
immersions from a possibly noncompact manifold and compute the Chern forms.
So we consider a holomorphic immersion
f: M - IP"'+\
where M is an m-dimensional complex manifold.
Remark. If M is compact and connected, then a theorem of Fulton-Hansen [FH]
implies that / has to be an embedding.
Recall that the (first) osculating map of f is given by
f^^): M - G(m+2,m+l) = 1?"^+^*, z h T^.
We will adhere to the following index ranges:
1 < i,j,k,«" < m; 0 < a,b,c,-" < m+1.
The Frenet bundle, 7 -♦ M, is a U(l)«U(m)>«U(l)-reduction of the pullback
bundle F U(m+2) -♦ M; if e = (e ) is a Frenet frame, then
[eo] = f, [eQA...AeJ = f<^).
Choose a Frenet frame e, and put
251

where fl = (fl*) is the skew-Hermitian Maurer-Cartan form of U(m+2). The
forms {ojf) satisfy
de = uf 9 e,.
a a b
The "Frenet conditions" imply that
0 m+1
Let ds^ denote the normalized Fubini-Study metric on IP'"+^ so that its
holomorphic sectional curvature equals 4 (and the volume of P*""^^ is ^r^"^ ), The
holomorphic immersion f pulls back ds to M giving it a KShler metric. The
forms (w') form a local type (1,0) unitary coframe on (M, f*ds ), and the
corresponding Kahler form is given by
Exterior differentiation of both sides of the equation J^ = 0 leads to
0
0^+^ A wj + ... + a/"+^ A a/? = 0.
1 U m 0
The holomorphy of f^ ^ is reflected by the fact that the forms (o/P"^ ) are all of
type (1,0). By Cartan's lemma
a/?+^ = Q..J
I ^ij 0
for some complex-valued local functions Q.. with Q.. = Q...
The type (2,0) symmetric form
II = a/?+^.wi = Q..wi. J
1 0 ^ij 0 0
is the complex second fundamental form of the immersion.
Theorem (The first normal form). Let f: M -♦ P*""^ be a holomorphic
immersion. Then in a neighborhood of any point in M there exists a Frenet
frame e such that
e*n?+^ = k.e*ni (no sum),
1 1 0 ^ '*
where the k.'s are globally defined real-valued functions on M with
0 < k < k ,<...< k,.
~ m - m—1 ~ "1
Proof. We want to see how the complex symmetric matric Q = (Q..) transforms
252

under a change of Frenet frame. Let e,e be two Frenet frames. Then on their
common domain the two frames are related by
e = e-g
for some U(l)«U(m)>«U(l)-valued local function g = (exp(tt), A, exp(tt)). Define
tilded quantities using e: u = e*n, 'X...uA = (J?"^^. From the formula
we compute that
(J = Ad(g ^)u = g ^a;g
Q = exp(-i(s+t))*AQA.
It now follows from a result of Chern (see [C] p. 28) that we can make Q
diagonal. The rest follows routinely. □
Given a holomorphic immersion f: M -» p*""^ we will call the global
functions
Kj = (k.)^: M - R
the complex principal curvatures. We also let a. denote the i-th elementary
symmetric polynomial of (k.). For example,
11 m' m 1 m
Let V denote the canonical connection (i.e., metric and type (1,0)) on the
holomorphic tangent bundle TM -♦ M coming from the KS.hler metric f*ds . In
the following we will compute the curvature matrix, Xt o^ ^ using the unitary
coframe (w'). We have
where 6 = (0*.) is the connection matrix.
Remaik. It goes without saying that the underlying Riemannian metric of f*ds
is quite special. For example, for m = 2, the o(4)-valued connection matrix of
M relative to (Reo'., Ioio'q, Re<*^, Imwj is computed to be
253

0
-i.;
Ke&l
Im(?J
i<
0
-Im&l
Re^
Ke0\
IraOl
0
-i^
-uel
Reel
'^
0
where 6 = ((?!) are the complex connection forms given earlier.
Coming back to our main discussion the curvature forms (x'.) are given by
x! = d^. + elh ^..
^i J k J
Using the Maurer-Cartan structure equations of U(m+2) we obtain
H = A '^ '^S - ^ '^ '^o = M - ^^') ^ <
j-O'
Thus
We easily have
J.= u}.- ^.ujI
J J J 0
k J k J
Again using the Maurer-Cartan structure equations of U(m+2) we obtain
J ' J
It follows that
j-O
m+l
Using the first normal form we can rewrite the above as
x! = wi A wi + <?! S o/J A ai + k.k. a/ A J.
'^j 0 0 J 0 0 1 J
We have, in particular,
trace x = -2i(m+l) $ - S k. wj A &J,
where $ is the KShler form of (M, f*ds ).
Note that if we let (h.) denote the holomorphic sectional curvatures relative
to the unitary coframe (w'), then
h. = 2(2 - «.).
The i-th Chern form of V, denoted by r.(M,V), is given by
254

TiCM.V) = P'(2^.x),
where P' denotes the i-th elementary symmetric polynomial in the eigenvalues of
the matrix j^'X- ^^^ example,
T-jCM.V) = ^ trace x,
-2(M.v) = (2i)'(.3/jx;: - xix^).
^m(M.') = (jf)"" let X.
We thus obtain
r^(M,V) = i (m+1) $ - Jp S «. (.; A &;.
Put
$ = i S 0/^+1 A 0;^+^
1 2 1 1
Observe that $. is the KS.hler form of the (possibly singular) osculating metric
f^ ■'*ds., where ds. denotes the standard metric on G(m+2,m+l). We can now
rewrite the preceding equation as
^l(M,V) = i ((m+l)$ - $j).
For m = 2 we calculate that
^2(m'.V) = i, (3 - (Tj + a^) $2.
For m = 3 we calculate that
^2(^''^) = ii» .^P2 - 3(«.+«.) + 2K.K.) *..,
where ^.. = uJl^JjJl^ uAmiA. Also
ij 0 0 0 0
^3(^''^) = sia (12 - 3(7j + 2(72 - 3^3) *'•
2 3
Theorem (Generalized Gauss-Bonnet-Chern). Let f: M -♦ P be a smooth
algebraic surface given by a holomorphic embedding, and also let c., c- denote
the Chern numbers of M. Then
(3a) c^ = 9d + i, f ((72 - 3(7^) ^\
(3b) c, = 3d + i, I (a, - a^) *^
where d is the degree of the projective variety f(M) c IP .
255

Proof. Wirtinger's theorem states that
The formulas now follow easily upon integration, a
If the complex principal curvatures coincide everywhere, i.e.,
we then say that the immersion f is totally complex-umbilic.
Proposition. The immersion f: M*" -♦ P'"'*" is totally complex-umbilic if and
only if the induced metric f*ds is KShler-Einstein.
Proof. A Kahler metric is KS.hler-Einstein iff the trace of the curvature matrix
is a multiple of the Kahler form. We saw that the curvature matrix, x* 0^ ^ds
is given by
trace x = -2i(m+l) $ - S k. wj A wj.
Thus trace x is a multiple of $ if and only if «=•••= « . a
Chern [C] has shown that a KS.hler-Einstein surface in P is locally either a
piece of a P or congruent to a piece of the normalized quadric
1110 ^
Q = {x +y +z +w = 0: x,y,z,w homogeneous coordinates} c P .
This result together with a bit of computation yields
Theorem. Suppose f; M -♦ P is a totally complex-umbilic immersion. Then
either « = «, = 0 (a piece of a P ),
or « = « = 1 (congruent to a piece of Q,).
We can give another characterization of the quadric in P as follows.
Theorem. Let M be compact and suppose that the osculating map
f^^); M - G(4,3)
is everywhere immersive. Then f(M) is a quadric in P .
Proof. Let V denote the canonical connection of M with the osculating metric,
i.e., the Kahler metric f^^)*dSj. The curvature forms, x' = (x'-). of (M,f^^)*dSj)
256

3 3
with respect to (w-.w-) are computed to be
X'l = -20.^ A 0.2 - 0^5 A 4 + o^O '^ ul
^'\ = -^1 = 0^5 A (.; + 0^3 A u\.
Let r.(M,V^), i = 1,2, denote the i-th Chern form of TM -♦ M with the
connection V . Consulting the above equations we find that
rjCM.V^) = i (3$j - *o)
Integration over M yields
•'M
s = <> + i> Lc^j - "i) *
A$ .
Combining these formulae with those in (3a,b) we obtain
SCg - Cj = 2d.
On the other hand we saw earlier in this section that
implying that
cj = d(4 - d)^ Cg = d(d^ - 4d + 6)
SCg - cj = 2(d - l)^d.
Consequently d = 2. □
In the following we write down the Gauss-Bonnet formulae for algebraic
3-folds in P^.
Theorem (Generalized Gauss-Bonnet). Let f; M -* ? be a smooth algebraic
3-fold given by a holomorphic embedding, and also let c., c.c., c be the Chern
numbers of M. Then
(4a) cj = 64d + ^J (-16(7j + Aa^ - a^) $^
(4b) CjC^ = 24d + i, j i-Sa^ + la^ - a^) $^
257

(4c) c, = 4d + i, Ij-a^ + }c^ - a,) *'.
where d is the degree of f(M) c P .
Proof, Wirtinger's theorem becomes
and the result follows upon integration, o
Given a holomorphically immersed hypersurface M*" -♦ p'"'*"^ we call
='=-- L'l^
the i-th total curvature. For algebraic hypersurfaces embedded in p*""*"^ we can
express the total curvatures in terms of the degree.
For algebraic surfaces of degree d in P , combining formulas in (la,b) with
those in (3a,b), we obtain
Kj = 2d(d - 1),
K^ = d(d - if.
For algebraic three-folds in P , we combine formulas in (2a-c) with those
in (4a-c) and obtain
Kj = 3d(d - 1),
K- = 3d(d - if,
K, = d(d - if.
2
Remark. The Chern form computations are often simplified using Todd's formula
which is of rather general nature. Consider a holomorphic immersion
f: M"" - P".
Associated to the map f are the Hermitian bundles T^*^ -♦ M. Todd's formula
enables one to compute the Chern forms of the tangent bundle TM -♦ M with
respect to the induced metric in terms of the induced KShler form and the Chern
forms of the Hermitian bundle T.^^ -♦ M:
258

r.(TM, f*ds2j = S^('"+^;+J)(ij$J) A r._.(T(l)),
where r. denotes the i-th Chern form and $ denotes the induced KShler form.
We point out, however, that the forms r.(T^ ^) are often difficult to compute.
§5. Surfaces in ps
Consider a holomorphic map
f: M - IP^
where M is a two-dimensional complex manifold. Throughout this section we
will assume that
(t) dim a(^) = 3.
The above condition means that the vectors
{dHldz^dz^, d'^{/d{z^f, d{^ld{z^f}
are linearly independent everywhere, hence we must also have
dim a(^) = 2.
It follows that the osculating order of f is
0(f) = 2,
and the osculating sequence of f is
(tj,t2) = (2,5).
From this we see that the holomorphic maps f and f^ ' are both immersive.
The Frenet bundle, 7 -♦ M, is a U(l)*U(2)>«U(3)-principal subbundle of
f" U(6) -♦ M. We fix a Frenet frame e = (e-,"-,e.) along f and write the
Chern forms and KsLhler form relative to it.
U(l)«U(2)xU(3) «^ 7 -4 U(6) *^ U(1)WU(5)
i i
M -4 IP8
259

We recall that the first Chern form of the line bundle t(°)* = T^H (H =
the hyperplane bundle on P ) on M is given by
TjCtC)') = -r,(T(»)) = i *.
where $ = * denotes the Kahler form of the induced metric on M by f
Similarly, the first Chern form of the vector bundle T^ ■' -♦ M is given by
where $. denotes the KEhler form of the first osculating metric ds., i.e., the
induced metric on M by f^^^: M -♦ G(6,3).
.(1)
S
1
i I
M -4 G(6,3)
Recall also that T^^^ = f^^^~^Sj, where S^ -♦ G(6,3) is the universal bundle.
Using Todd's formula we find that
(a) rj(TM,f*ds2) = i(3$ - $j).
Similarly
(b) T^iTUfds'^) = i,(3$2 _ 2$A$j) + TjCT^l)).
Let V^ ^ denote the type (1,0) metric connection on the Hermitian bundle
Weyl's formula then gives
r^i^^^h = i(-82* + (Si+Sj)^! - Si$2) = ^-^^ + 5$j - 2$2).
But ^2 = ^ ""c® ^^^•' ^ -* G(6,6). It follows that
(*) rj(v(l)) = i(-3$ + 5$j).
Computations (somewhat tedious) also show
r2(v(^b = ^j(6^^ - 14$A$j + 11$J) + r2(T(^)).
Observe that with the assumption (f) on f we have isomorphisms
260

a(*) n t(°) • S'TM,
aW* 9 a('+^) ^ (S'TM)* • S'+^TM,
where S means the b-th symmetric power. With these isomorphisms in mind
further computations with the Chern forms reveal
(c) 3rj(TM,ds2) = i(-3$ + 5$^),
(d) llTjCTM.daJ) + rJ(TM,dsJ) = i,(6$^ - 14$A$j + 11$J) + t^{T^^\
where ds- denotes the osculating metric.
Assuming that M is compact (and f is an embedding) we have
Cj(M) = [rj(TM,f*d82)] = [TjCTM.ds^)],
where [•] denotes the cohomology class. It follows that
(A) Cj(M) = |i*J = [jI*].
Substitution of (A) into (b) and (d) yields
(B) ^(M) = [^,$2j
Assume that f: M -* P is a holomorphic embedding from a connected
compact M. We then have
d =r deg f(M) c IP^
dj = deg f^^)(M) c P^, N+1 = (J),
c., Cj = the Chern numbers.
We also put
•'M
The numbers (. are called the polar classes of M in P .
We now have
9d = 4dj,
cj = 9d - 6^j + dj,
C2 = 3d - 2^j + ^2,
261

9cJ = 9d - 30^j + 25dj,
llCj + cj = 6d - 14^^ + lldj + ^2-
The first of the above formulae, for example, follows from (A) and Wirtinger's
theorem
Corollary. Let f: M -* P be an algebraic surface satisfying the condition (f).
Then
Cj = dj = jd.
In particular, 4 divides d, 9 divides c. and d , 3 divides c. and („, and 6 divides
^1-
A Veronese surface in P has
d = 4, cj = dj = 9, Cj = ^2 = 3' ^1 = ^•
Only other nondegenerate surface in P with degree 4 is a rational normal scroll
whose osculating order is 3 with osculating sequence (2,4,5). (Recall that the
minimal degree of a nondegenerate surface M in P is codim(M) + 1 = 4.)
Theorem ([T]), Let f: M -♦ P be a holomorphic embedding from a (connected,
as always) compact surface. Then f(M) is a Veronese surface.
Proof. We know that
x(V = i-q + Pg.
where
q = dim H (M,^ ) = the irregularity of M,
p = dim h\m,0^) = dim H°(M,<7(K)) = the geometric genus.
Suppose H (M,^(K)) ^ 0. Then the canonical linear system |K| contains an
effective divisor D. We have D-H (the intersection pairing) > 0, where H
262

denotes the hyperplane class. But
D-H = Cj(K) U Cj(H) = -Cj(M) U Cj(H)
which is a contradiction. So, p =0 and x(^m) = 1 - q < 1. But Noether's
formula together with Corollary give
hence
and finally d. must equal four, o
One speculates whether or not the rational normal scroll in IP is the only
algebraic surface whose osculating sequence is (2,4,5).
For the rest of this section we will give a Chern form computation proving,
in particular, (c). Let M be any Hermitian complex surface and also let TM -♦
M denote its holomorphic tangent bundle. We fix a local unitary frame, e =
(e.), of TM -♦ M, and compute the Ghern forms relative to it. We will use the
following index conventions:
1 < i,j,k < 2; 3 < a,b,c < 5; 1 < o,/?,7 < 5.
Recall
de. = or • e.,
where (uj\) are the connection forms of the bundle TM -♦ M. Also
doj\ = - uj} h UK + y!,
J k J ^J'
where (x') are the curvature forms of TM -♦ M
Consider the dual bundle T M -♦ M. Letting (^) denote the dual type
(1,0) coframe we have
From this we see that
de* = - d A ^.
J
X(T*M) = -x(TM).
263

Consequently,
rj(T*M) = - rj(TM),
where r. denotes the first Chern form.
2
Let S TM denote the second symmetric power of TM. We take as a local
unitary frame
Ej = ej«ej, Eg = e^9e^, E3 = {i/^/2){e^9e^ + e^9e^).
Then the induced connection matrix relative to (E ) is given by
(*)
2w} 0 ^/2uJ^
0 2a^ ^/2(J?^
Proof. For example,
dE = de. • e. + e, • de
= w: • e. • e. + 0^ • eg • e^ + e^ • Wj • e. + e. • Wj • e,
= 2w} • Ej + ^/5a^ • E^.
This verifies the first column of (*). The rest is similar, a
2
From (*) we compute the curvature forms of S TM -♦ M.
2
Lemma. Let x denote the curvature matrix of S TM relative to the connection
matrix in (*). Then
Xj = 2Xj, X2 - 2X2. X3 = Xj + X2.
Xi = 0, x5 = ^/5xJ, X2 = ^/5x^•
Proof. Let fJ denote the matrix in (*). Then
X = dn + n A n.
For example,
x\ = 2dw} + 0 + 0 + ^/5w2 A ^/2(J?^ = 2xJ. □
264

It follows at once that
rj(S^TM) = 3.rj(TM).
Consider the Hermitian bundle T*M«S^TM -♦ M. Put
Then (cj«cj form a unitary frame for T*M«S^TM -♦ M. We let (fiij) denote
the corresponding connection forms. So, for example,
On the other hand,
d(Cj« Cj) = d(?^ • Ej + ^ • dEj
= wj • (ej« €3) - wj • {e^9 63) + y/iul • (fj* 65),
and
nl3 1 n23 .1 nl5 /r 2
Similar calculations yield the connection matrix (n'.*). Let (x-u) denote the
corresponding curvature forms, i.e..
"4* = ^'^ + "ic * n'b-
We compute that
yl3 = yl yl4 ^ 2y2 _ yl yl5 ^ y2
y" = 2y^ - y2 y^^ = y^ v25 = yl
'^23 ^1 ^2' ^24 ^2' ^25 ^l'
It follows that
r^(T*M«s2TM) = ^^ S xj: = fi-CxJ + x',) = 3.r^(TM).
This proves (c).
We could have used the following general formula to establish the last step
of the above derivation: Let E, F -♦ M be Hermitian bundles of rank p, q
respectively. Also let V_, V_ denote the connection matrices of E and F (fix
unitary frames), respectively. Then
where V-,-,_ is the induced connection matrix of the tensor bundle E • F, and I
265

denotes the identity matrix.
Letting E = T*M and F = S^TM we then have
E®F
-"^1
A
A'
A.
• I3 + I2 •
■ 2wJ 0
0 20,2
/T^-
So, for example,
n}^ = S (l,l)-entry « (l,l)-entry = -wj + 2w} = w},
n|j = S (1,1) -entry « (3,l)-entry = -wJ-0 + ^^/5w^,
nJ5 = S (2,l)-entry * (l,l)-entry = -u}\.
Notice that
V* = -*V,
where V is the connection matrix of TM -♦ M, and V*, its dual.
266

Appendix I. Some Background on
the Linear Algebra of Complex Forms
Robert Fisher
§1. Complexification
What follows is somewhat of a pedagogical discussion of the process known
as "complexifying a real vector space". Some of the topics which are related to
complexification, and which enable one to skillfully use the graded algebra of
complex forms that accompanies any complex manifold will also be presented.
The complexification of E, which will be denoted by E_, is defined as follows:
(1.1) E^ = E X E.
In a natural way, the complexification E^ can be viewed as a complex vector
space. Explicitly, the complex scalar multiplication is defined as follows:
(1.2) (a+ib)-(u,v) = (au-bv, av+bu),
where (a+ib) € C and (u,v) € E . It is evident that this scalar multiplication is
done by direct analogy with the usual multiplication of complex numbers. In any
case it is in this manner that E^^ is viewed as a complex vector space. Moreover,
it is elementary to show that if E has real dimension n, then as a complex vector
space E^ also has dimension n. Finally we point out that it is customary to
write u+iv instead of (u,v). Part of the motivation for this stems from the
desire to identify E with the "x-axis" inside E_. In other words, u € E is
identified with the vector (u,0) € E^. With this convention the definition of
complex scalar multiplication is consistent with iv corresponding to (0,v) € E_.
For the moment we move in a different direction and consider a natural
companion to this idea of complexification, namely the idea of a complex vector
space structure. Explicitly we have the following: a complex vector space
structure on the real vector space E is a real linear map
267

(1.3) J: E - E
such that JoJ = -id. More simply, one writes J = -id. When E is a finite
dimensional vector space, an elementary exercise shows that the existence of a J
is equivalent to the requirement that E be even dimensional. Continuing, let J
be a complex structure on E. Then with respect to J, there is a complex scalar
multiplication on E. It is defined as follows:
(1.4) (a+ib)-u = au + bJ(u),
where (a+ib) € C and u € E. With this definition the pair (E,J) is a complex
vector space. If the dimension of E is 2n, then the complex dimension of (E,J) is
n. Still assuming that dim(E) = 2n, we say that a real basis (u.,'",u, ) for E
is adapted to J provided that
J(u.) = u ,., J(u J.) = -u., 1 < j < n.
An elementary linear algebra exercise shows that such bases clearly exist.
Finally, if E is already a complex vector space, then when viewed as a real
vector space E has a canonical J, namely, the one defined by
J(u) = iu.
In general an even dimensional real vector space E has many complex
vector space structures. Indeed it can be shown that the real general linear
group of E, denoted by G = GL(E) (so G = GL(2n,IR) upon choosing a basis),
acts transitively on the left on the set S of all complex vector space structrues on
E. The action is given by
A'J = AoJoA" ,
where A € G, J € 5. Fixing a J € <S, which will serve as a base point, the
isotropy at J, i.e., the subgroup H = {A € G: AoJ = Jo A}, is a closed Lie
subgroup of G. In fact, H is nothing but the complex general linear group
GL(n,C) upon choosing a basis. The corresponding coset space
G/H ^ GL(2n,IR)/GL(n,C)
268

is then identified with <S in a canonical fashion. Thus, after a choice of base
point, the set S is viewed naturally as a homogeneous space in the traditional
sense.
Next, we give the relationship between the two concepts introduced above.
So let E be a real vector space and let J be a complex vector space structure on
it. Since (E,J) is already a complex vector space, it is then natural to ask the
following question: why bother to complexify a complex vector space? To answer
this question observe that J has two eigenvalues, namely ±i. So simply put, the
complexification of E is a natural vector space in which the eigenspaces of J
reside naturally as complex subspaces. More explicitly, we proceed as follows:
there is a canonical complex linear extension of the given J to E_. It is the map
J^: E^ -♦ E^ given by
(1.5) Jc(u+iv) = J(u) + iJ(v).
Traditionally no notational distinction is made between J and J^. As long as no
confusion is likely we will follow this tradition. No matter, the complex splitting
of E_ determined by J goes as follows: define
(1.6) E^'° = {(u-iJu): u € E}, E°'^ = {(u+iJu): u € E}.
Th following lemma is an elementary exercise in linear algebra:
Lemma 1. E ' is the (+i)-eigenspace of J, E ' is the (-i)-eigenspace of J, and
E^ = E^'° ® E°'^
c
is a splitting of E_ into complex subspaces.
We point out that in the course of proving the lemma the reader will
observe that (E,J) is isomorphic as a complex vector space to E ' under the
map
(1.7) u H u - iJu,
and it is C-anti-isomorphic to E ' via the map
(1.8) u H u + iJu.
269

Thus E ' and E ' are not isomorphic as complex vector spaces, but only
C-anti-isomorphic under the canonical conjugation map on E_ given by
u + iv H u - iv.
Duality and Complexification
In order to work successfully with complex forms on a complex manifold it
is important to take stock of the formal distinction between two concepts: firstly
there is the complex dual space of the complexification of E, and secondly there
is the complexification of the real dual space of E. To begin, let (E_)* denote
the complex dual space of E^ and also let E' denote the real dual space of E.
Consider the following C-linear map C: (E')^-, -♦ (E_) given by
(1.9) C(fj+if2)(u+iv) = (fjCuH^Cv)) + i(fi(v)+f2(u)),
where fj+if, € (E')^^ and u+iv € E^^. Evidently this map is injective and thus
when E is finite dimensional (1.9) is a complex vector space isomorphism.
Suppose next that we are given a complex vector space structure J on a
real vector space E. In a natrual way J determines a complex vector space
structure on the real dual space E'. Namely we have J': E' -♦ E' given by
(1.10) J'(f)(u) = f(Ju)
for f € E' and any u € E. Thus by analogy with the above discussion the pair
(E',J') is a complex vector space. Next, by applying Lemma 1 we have the
eigenspace splitting of (E')_ determined by J', namely,
(1.11) (E')c = (E')^'° ® (E')°'^
Viewing (1.11) in terms of the map C in (1.9) we have the following:
Lemma 2. Assuming that E is a finite dimensional it follows from (1.9) that
C((E') ' ) is the complex subspace of (E_)* which is characterized by the
property that E ' is contained in the kernel of each one of its elements.
Analogously the image of (E') ' under C is characterized by the property that
each of its elements annihilate E ' .
270

The proof is elementary and thus is left to the reader.
§2. Complex Forms
The goal of this section is to apply the concepts introduced in the previous
section to the algebra of complex forms that accompanies any complex manifold.
So let (M,J) be an almost complex manifold. Recall that this means that M is a
smooth even dimensional real manifold and that J is a global endomorphism of
the tangent bundle with the property that J = -id. (Note that J defines an
orientation on M.) Said differently, J is a smooth section of the vector bundle
L(T(M)) = T*M • TM with the property that at each x € M the value of the
section J is a complex vector structure on T M. The section J is traditionally
called an almost complex structure for M. Now then, any complex manifold has
a canonical almost complex structure. In fact, an almost complex manifold (M,J)
is a complex manifold if and only if the torsioh tensor field of J vanishes. For
more information on this important point see chapter 9 of [KN]. For the current
purpose the only necessary assumption is for (M,J) to be an almost complex
manifold. The stronger condition on J is not required.
To begin, consider the complexified tangent bundle TM_. Explicitly this
bundle is the Whitney sum of the tangent bundle with itself, that is,
(2.1) TM^ = TM ® TM.
This expression gives the bundle construction which is analogous to (1-1). In
keeping with the discussion in section 1 we will write (2.1) as
(2.2) TM^ = TM + iTM.
In this description the tangent bundle of M is identified to the obvious real
subbundle of TM^^. For emphasis we point out that TM^ is a complex vector
bundle. It is evident by appealing to (1.2) that each fibre (TM_) is a complex
vector space.
271

Simply put, the complex forms of degree k are the smooth sections of
A (TM^) , the kth exterior power of the complex dual of TM^. It is the custom
to identify A (TM^) to the vector bundle whose fibre at a point x € M is the
complex vector space of alternating k-linear maps from the k-fold Cartesian
product of (TM^) , denoted by T M-, to C. With this identification, a complex
k-form is a smooth section w of A (TM^) whose value at x € M is an
alternating k-linear map u{x): T M^ -♦ C.
Let XJiM) denote the complex vector fields on M, i.e., the smooth sections
of the complexified tangent bundle TM^. From (2,1) it follows that XJM) is
canonically the complexification of the real vector fields on M, namely X{M),
thereby justifying our notation for the complex vector fields. Next let C"'(M)
denote the ring of complex valued functions on M. Due to the fact that the
complex k-forms are tensors any complex k-form u can be identified canonically
with an alternating C"'(M)-linear map
(2.3) ^^(M)'^ - C"(M),
where 'l'-^(M) denotes the k-fold cartesian product of XJM). Many of the
arguments involving complex k-forms, which are global in nature, are presented
more cleanly by dealing with this common description of complex k-forms. We
will also proceed in this manner tacitly using the identification given by (2.3).
The inexperienced reader is urged to consult Theorem 2 on page 162 in [S] for a
more thorough discussion of (2.3). Either way we will use the symbol A (M) to
denote the vector space of all complex k-forms. Juxtaposed to A (M) is the
vector space of all real valued k-forms. They are described by analogy with the
complex case and will be denoted by A_(M) so as to distinguish them from the
complex case. Contrasting these two infinite dimensional vector spaces we
observe the following:
k k
Lemma 3. A (M) is canonically isomorphic to the complexification of Ap(M).
272

Proof. From the fact that the complex vector fields are naturally seen to be the
complexification of the real vector fields, that is,
-r^(M) = ^(M) ® ^(M),
it is easily observed that the procedure outlined by (1.9) in the previous section
establishes the lemma. To be a bit more detailed let
(2.4) C: k\{U)^ - k\u)
denote the map constructed by direct analogy with (1.9). Next recall that in
Lemma 2 the vector space is assumed to be finite dimensional. Anyone actually
trying to prove this lemma will realize that the finite dimensionality of E
guarantees the surjectivity of the map C since its injectivity is evident. In the
present situation the relevant vector spaces are all infinite dimensional and so in
principle sthe surjectivity of (2.4) is an issue. However, because one is dealing
with vector spaces which are the section spaces of vector bundles the argument
behaves as if one is in the finite dimensional situation. Thus, given any u) €
A (M) one uses a fibrewise construction to obtain a candidate which is formally
mapped to u) under C. That this candidate is well-defined, i.e., smooth, can then
be done easily by a local argument.
For the purpose of demonstration and later reference we specifically do the
k = 2 case. The map
(2.5) C: a2(M)(, - k\M)
is defined as follows:
C{u)^+iu^){Z^,Z^) = {[a;^(X^,X2)-a;^(Y^,Y2)] - Wp.^,Y^)+ujp^;:f.^)]}
+ \{[iJ^{:^^,\)-u)p^;Y^)] + [a;^(X^,Y2)+a;^(Y^,X2)]},
where Z. = X.+iY. € XJIA), This is not an aesthetically pleasing display of
information. However, if one looks a bit closer, then it becomes apparent that
(2.6) u) = C{u}^ + iWj) = C{u}^) + \C{u}^).
Also one can then verify that the real and imaginary parts of (2.6) are given by
273

(2.7) Re(a;) = ReC{tJ^) - lmC{uj^), Im(a;) = Ree{uj^) + ImC{uj^).
With this information it is not hard to show by hand that the map C is both
injective and surjective.
The next phase of the discussion involves the interaction of the almost
complex structure with TM^^. Firstly with respect to the given almost complex
structure J, the tangent bundle TM becomes a complex vector bundle whose
complex rank is one half of the rank of TM as a real vector bundle. By analogy
with section 1, J extends to a C-linear bundle map on the complexified tangent
bundle TM,,. Consequently the vector bundle analogue of Lemma 1 holds,
namely, each fibre of TM,, splits by Lemma 1 into the ±i-eigenspaces of J at x:
T M^ = T^'°M ® T°'^M.
X C X X
Thus there is a splitting of TM as a direct sum of complex subbundles:
TM^, = T^'°M ® T°'^M.
Remark. (TM,J) is isomorphic as a complex vector bundle to T ' M under the
bundle map
X H I (X - iJX)
for X € T M, and it is C-anti-isomorphic to T ' M under the map
X H i (X + iJX).
While the factor ^ is something of a nuisances, it appears quite naturally when
one expresses a complex tangent vector in terms of its (1,0) and (0,1)
components. Namely, if Z € T^M^_,, then Z = Z^'° + Z°'^ with
Z^'° = I (Z - iJZ) € T^'°M,
Z°'^ = I (Z + iJZ) € T°'^M.
Finally when (M,J) is a complex manifold (TM,J) and the eigensubbundles
T^'^'m and T ' M are all holomorphic vector bundles. Of course, in this
situation (TM,J) is isomorphic to T ' M under the above map as a holomorphic
vector bundle. For this reason T ' M is often refered to as the holomorphjic
274

tangent bundle. For completeness sake we note that T ' M is not isomorphic to
(TM,J) as a holomorphic bundle. It is only anti-isomorphic.
Next we observe that the information globalizes to the vector space of real
vector fields X{M). The almost complex structure determines a complex vector
space structure on X{M). Traditionally one uses the symbol J to indicate both
the almost complex structure on M and the complex vector space structure on
<r(M). Thus by (1.4) the pair (<l'(M),J) is a complex vector space. Continuing,
we follow by analogy with (E',J') in (1.10) and observe that J also determines a
complex vector space structure on the real vector space A_(M). The desired
complex vector space structure is given as follows:
(2.8) J'a<X^,...,X^) = J*a<Xi,---,X^) = a<JXj,. • .,JX^),
where w € A (M) and X, € ^(M). Said differently, the complex vector space
structure J' is the pullback by J. Applying Lemma 1 to the case (-?(M),J) and
(A (M),J') we have the following:
Lemma 4. The complex vector space 'l'p(M) and A'(M)j^ are expressed as the
direct sum of the ±i-eigenspace8 of J and J' respectively. Explicitly one writes
X^{M) = ^^'° ® /'\
^r(^)c = Ar(M)^'° e A^(M)0'^
Note that X ' (resp., -r' ) is canonically isomorphic to the sections of
T^'°M (resp., T°'^M).
The last part of our discussion centers around three things: the
interpretation of Lemma 4 in terms of (2.4), the notion of complex forms of
complex type (p,q), and some special statements about complex forms of type
(1,1). To this end we first note that the analogue of Lemma 2 holds for
(A^(M),J'). Explicitly from (2.4) let
(2.9) A^'° = C(A^(M)^'°), A°'^ = C(A^(M)°'^).
These complex subspaces of A (M) admit the following alternate descriptions:
275

(2.10) A^'° = ^"^ ^ A^(M): 'XZ) = 0, Z € Z'^},
A°'^ = {w € A^(M): a<Z) = 0, Z € /'"}.
Traditionally the complex subspaces A ' and A '" are called the complex
1-forms of type (1,0) and (0,1) respectively. Evidently this notion of typing is
dependent upon J.
Next, we note that there is a canonically defined conjugation map on
A (M), Via (2,4) it is compatible with the obvious conjugation map on A*(M)j^,
Explicitly on complex 1-forms the conjugation map is given by
(2.11) u{Z) = a<Z),
where Z = X-iY for Z = X+iY € X„{M) and the second horizontal line on the
right side of (2.11) denotes ordinary complex conjugation in C. The
generalization of (2.11) to complex k-forms is straightforward. In any case under
this conjugation map the spaces A ' and A ' are C-anti-isomorphic to each
other.
Let p,q be a pair of nonnegative integers with 0 < p+q = k < 2n. Using
the complex analogue of the wedge product (cf. [AMR] p. 392) we define the
complex k-forms of type (p,q) to be the complex submodule A^'*^ generated by
the decomposible tensors
W, A'"AW A^ A'«'Afi^,
1 P
where w. € A^'° and ^ € A°'\ From the splitting of ^^-.(M) and (2.10), it
follows immediately that
(2.12) u) € AP''^ if and only if u{Z^,' • -.Z ^ ) = 0
whenever the complex vector fields Z. are pure either of type (1,0) or type (0,1),
and more than p of the Z, are of type (1,0) or more thatn q of the Z. are of type
(0,1).
Summarizing A (M) is the following direct sum of complex subspaces:
276

(2.13) a''(M) = ® AP'^ A'l'P = AP'^i.
0<P,q;p+q=k
Collectively
(2.14) A = ® a''(M) = ® AP'^i
k<0<2n
along with the wedge product of forms is the algebra of complex forms. Note
that A (M) = C"'(M). The first decomposition in (2.14) is canonical and the
second is determined by the almost complex structure J on M.
§3. Complex (l,l)-forms
In this section the information of section 2 is specialized to the case of
complex 2-forms of type (1,1), thereby extracting some useful expressions of this
menagerie of data. To begin recall (2,7), We will say that a real 2-form (or
more generally, a real k-form) u is J-invariant whenever
(3.1) J'a;= J*a;= w.
In other words, for each pair of real vector fields X,Y,
a<JX,JY) = a<X,Y).
2 2
To establish notation let A.(M) denote the compelx subspace of (A (M),J')
which consists of all J-invariant real 2-forms. Note that A (M) is always
nonzero. Indeed the map
w H i (w + J*a;)
2 2
is an idempotent of A (M) onto Aj(M). Reexamining equation (2.5) it is fairly
2 11
immediate that A (M)-, is mapped isomorphically to A ' by C, Thus one sees
that a complex 2-form of type (1,1) is viewed naturally as a pair of J-invariant
real 2-forms,
Let u,v € /f ' , Each is expressed uniquely as
u = X - iJX, V = Y - iJY
for real vector fields X,Y, We specialize equation (2.5) further as follows: given
277

any pair w. (j = 1,2) of J-invariant real 2-forms let
w = C{u^ + iWg).
Evaluating u on the pair (u,v) we get
(3.2) a<u,v) = 2(a;^(X,Y)-a;2(X,JY)) + 2i(a;^(X,JY)+a;2(X,Y)).
In particular, if w = C{u.) (i.e., w. = 0), then (3.2) reduces to
(3.3) a<u,v) = 2(a;^(X,Y) + ia;2(X,JY)).
In turn by evaluating (3.3) on pairs (u,a) the expression simplifies nicely to
(3.4) a<u,a) = 2ia;j(X,JX).
One has the analogous expressions when w = iC{u)„), namely
(3.5) a<u,u) = -2a;2(X,JX).
In any case the above specialization leads to a piece of common terminology! let
w be a complex 2-form of type (1,1). One says that w is a real (l,l)-form
provided the following condition is satisfied:
(3.6) u = e{uj)
for some J-invariant real 2-form u. The (l,l)-form w is called real due to the
isomorphism given in (2.4). Taking (3.4) into account one obtains an additional
reason why this is a proper use of the language. Indeed if one multiplies by i,
then
(3.7) ia<u,a) = 2a<X,JX)
which is genuinely a smooth real valued function on M for each vector field u of
type (1,0).
The notion of a positive real (l,l)-form is a further refinement of (3.6).
Simply put, a real (l,l)-form u is positive provided that
(3.8) i£j(u,a) is a positive function
for each nonzero vector field n € X ' .
278

References
[AMR] R. Abraham, J. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and
Applications, Springer-Verlag, 1988,
[KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol 2,
John Wiley and Sons, 1969.
[S] M. Spivak, Differential Geometry, Vol. 1, Publish or Perish, 1979.
279

Appendix II. Elliptic Functions
By a lattice in C we mean a rank 2 free abelian subgroup of C which
generates'C over the reals. If (t'jiti^o ^'^^^ ^ ^^^^^ ^^ ^ lattice L, then we write
L = <tj^, u}^> c C.
Without loss of generality we assume that Im(a;./a;,) > 0 so that r = '*'i/S ^^®^
in the upper half plane
H = {x+iy € C: y > 0}.
Suppose we have another basis Wj.w, °^ ^ ^^*^ ^ — ^\l^2 ^ ^' ^® ^^^^
have
where ad - be = 1, and a,b,c,d € ff. Conversely, given any A € SL(2,ff)
(*) \^-) = A-Vi)
is a basis of L with r € H.
The action (*) induces an action of SL(2,ff) on H:
ar + b
r H
cr + d
for r € H, [*'j] € SL(2,ff). It is easy to see that the diagonal matrices {±1} act
trivially on H so that we have an action of PSL(2,ff) = SL(2,ff)/{±I} on H.
Consider the subset c H
D = {z € H: -1/2 < Re z < 1/2, |z| > 1}.
It is routinely verified that D is a fundamental domain of the action of PSL(2,2?)
on H, i.e., every orbit is represented in D and that two points of D are in the
same orbit if and only if they lie on the boundary of D.
Two lattices L^.L, C C are said to be isomorphic to each other if
oL. = L, for some o € C.
The tori C/L , C/L, are said to be isomorphic if their lattices are. A torus C/L
is made into a complex manifold by requiring the projection r: i -* C/L be
280

holomorphic. Only holomorphic maps between complex tori are group
homomorphisms composed with translations. We thus have
Proposition. The tori C/L. and C/L are biholomorphic if and only if they are
isomorphic.
Consider the bijective correspondence
{isomorphism classes of tori} -» H/PSL(2,ff),
taking the class of C/L to the class of r € H. Later we will define so called the
modular function
H/PSL(2,ff) -* C.
The modular function is a bijection thereby parametrizing the totality of complex
tori by C.
To explain and motivate the aforementioned modular function we now
briefly review the theory of elliptic functions.
Fix a lattice L = <u., w >, u)Ju)„ € H. An elliptic function f relative to
the lattice L is a meromorphic function on C which is L-periodic, i.e.,
f(z+w) = f(z), w € L.
Thus an elliptic function relative to L may be thought of as a holomorphic
function
f: C/L -* C U {od} = P^
Conversely, any holomorphic function C/L -» P arises as an L-periodic
meromorphic function €. (It is customary to assume that the f(€/L) ^ {od}.)
A fundamental parallelogram for the lattice L is the set
{a + tjWj + t^u^. 0 < t < 1},
where a is a fixed complex number. A fundamental parallelogram is naturally
identified with C/L. Unless otherwise specified we shall work with the standard
fundamenatal parallelogram given by a = 0.
We have the following well-known results on elliptic functions.
281

Theorem A. Let P be a funcdamental parallelogram and assume that the elliptic
function f has no poles on the boundary dP. Then the total residue of f in P is
zero.
Proof. We have
25ri S Res f = I fdz = 0,
where the last equality being valid due to the periodicity (the integrals on
opposite sides cancel each other), o
Theorem A says that the total residue of a meromorphic function on C/L
(i.e., a holomorphic function C/L -» P ) must vanish. Note that Theorem A
generalizes to the following result: the total residue of a meromorphic 1-form on
any compact Riemann surface must vanish. Note also that as an immediate
corollary to the above theorem we have: an elliptic function must have at least 2
poles counting multiplicities.
Theorem B. Let P be a fundamental parallelogram, and assume that the elliptic
function f has no zero or pole on 5P, Then the sum, S a,, of the orders of zeros
and poles of f inside P is zero.
Proof. Since f is elliptic, so are f and f'/f. Now
0 = [ f'dz/f = 2;ri S a.
by the argument principle, o
Let {p.}"_i C P be the set of zeros and poles of f with multiplicities {a.}.
The divisor (f) = S a.p. on C/L is called the divisor of f, and S a. is the degree.
Theorem B generalizes to the following: the divisor of any meromorphic function
(i.e., a principal divisor) on any compact Riemann surface has degree 0.
Theorem C. Maintaining the preceding notation we have
a.Pj + • • • + a p =0 (modulo L)
where "+" denotes the complex addition. In other words, if E a.p. = (f) is a
282

I
principal divisor on C/L, then
in C/L, where "+" denotes the group addition in C/L.
Proof. Observe that
zf'(z)/f(z) dz = 25ri (a^pj + ••• + a^p^)
since Res (zf'(z)/f(z)) = a.p.. On the other hand we can compute the integral
over the boundary of the parallelogram by taking it for two opposite sides at a
time. For example,
can be computed by letting u = z - w in the second integral. We then obtain
-Wg ^'(u)/^(u) du = 2mnuj
"'a
for some integer m. The rest follows, a
Abel's theorem says that any two integral divisors of degree d on C/L are
linearly equivalent if and only if their respective group sums coincide. Theorem
C, then, reflects the fact that a principal divisor is linearly equivalent to the
trivial divisor.
We now establish the existence of elliptic functions by defining the
Wderstrass p-function. Consider the infinite series
p(^) = ;^ + M^. - U.
where the sum is taken over all nonzero lattice points w € L\{0}. Note that the
sum
S l/|a;|\ we L\{0}
converges for X > 2. It follows that the above infinite series coverges uniformly
on compact subsets of C away from the lattice points. This proves that p(z) is a
meromorphic function on C.
283

The series expansion of p(z) shows that it has a double pole at the lattice
points, and no other. Easily p{z) = p(-z).
Differentiating p(z) term by term we obtain
where the sum is taken over all lattice points a; € L. Note that p' is odd, i.e.,
p'(z) = -p'(-z). It is also clear that p' is L-periodic. From the periodicity of
p' we see that
PCz+Wj) = p(z) + c
for some constant c € C. It follows that
p(a;j/2) = p(-a;j/2) + c,
and since p is even we must have c = 0. This argument shows that p is
L-periodic, which was hard to see directly from the series expansion of p.
The set of all elliptic functions relative to L forms a field, and is called the
rational function field of C/L.
Theorem D. The rational function field of C/L is C(p,ji'). In other words, any
elliptic function relative to L can be written as the ratio of two complex
polynomials in p and p^
Proof. An elliptic function f can be written as a sum of an even and an odd
function:
f(z) = i [f(z) + f(-z)] + i [f(z) - f(-^)].
If f is odd, then the product fp' is even. Thus it suffices to prove that C(p) is
the field of even elliptic functions, i.e., if f is even, then f is a rational function of
p. Suppose that f is even and has a zero of order m at some point u. Then f
also has a zero of the same order at -u since
similarly for poles. We have
Claim. If u = -u (mod L), then f has a zero or a pole of even order at u,
284

Let us assume the claim and proceed with the proof. Let u., 1 < i < r, be
a family of points containing one representative from each class (u,-u)(mod L)
where f has a pole or a zero, other than the class of L itself. Let
m. = ord f if 2u. ^ 0 (mod L),
m. = 5 ord f if 2u. = 0 (mod L).
From the above remark we know that for a € C, a ^ 0 (mod L), the function
p(z) - p(a)
has a zero of order 2 at a iff 2a = 0 (mod L), and has distinct zeros of order 1 at
a and -a otherwise. Hence for all z ^ 0 (mod L) the function
n [p(z) - p(u.)]'"^
has the same order at z as f, This is also true at the origin because of Theorem
B applied to f and the above product. The quotient of the above product by f is
an elliptic function without zero or pole, hence a constant, o
Proof of Claim. Note that u = -u (mod L) is equivalent to
2u = 0 (mod L).
On C/L there are exactly 4 points with this property, namely
0, Wj/2, W2/2, {u}^+u}^)l2.
If f is even, then {' is odd, i.e., f'(u) = -f'(-u). Since u = -u (mod L) and f is
periodic, it follows that f'(u) = 0 so that f has a zero of order at least 2 at u.
If u ^ 0 (mod L), then the above argument shows that
g(z) = p(z) - p(u)
has a zero of order at least 2 (hence exactly 2 by Theorem B and the fact that p
has only one pole of order 2 on C/L). Then f/g is even, elliptic, holomorphic at
u. If f(u)/g(u) ^ 0 then ord f = 2. If f(u)/g(u) = 0 then f/g again has a zero of
order at least 2 at u and we can repeat the argument. If u = 0 (mod L) we use
g = 1/p and argue similarly, proving that f has a zero of even order at u. o
We shall now write down the power series expansion of p and p' at the
285

origin,
pW = :,+ 5:ii;,(i + i + (i)'+•••)'-ij
= i, + S 1 (m+lXi)"-1, (w e L\{0})
W m=l
OD
= 72 + S C^z"",
—1 ^
m!=l
where
c = S (m+l)/a/"+2.
Note that c = 0 if m is odd.
m
Put
we can write
s (L) = s = S f-
p(z) = i, + S (2n+l)s2„./"
n=l
= ij + 3s^z^ + 5Sgz'* + •.. ,
Differentiating term by term we obtain
P'(2) = jl + ^^4^ + 20SgZ^ + • • • ,
A consequence of these calculations is
Theorem E. Let ggCL) = gj = SOs^, and gjCL) = gj = UOSg. Then
p'2 = 4p^ - g^p - gj.
Consider the cubic polynomial equation
y^ = 4x' - g^x - gg, g. = g.(L).
The discriminant of the right hand side of the above equation is given by
A(L) = A = g^ - 27g2.
Put Wj = {u^+uj^)/2, and
e. = p(a;./2), 1 < i < 3.
Then the function f(z) = p(z)-e. has a zero at oj./2 which is of even order so that
p'(w./2) = 0. Comparing zeros and poles we see that
286

p'2(z) = 4(Kz) - ei)(p(z) - e2)(p(z) - e^)
so that the polynomial 4x -g-^-g, has 3 distinct zeros and A # 0. (p takes on
the value e. with multiplicity 2 and has only one pole of order 2 mod L, hence
the e.'s are distinct.)
Consider the map
$: C/L - P^ z M H [l,p(z),p'(z)], 0 H [0,0,1].
(Keep in mind that we often do not distinguish z € C with t(z) € C/L.) The
precedog discussion tells us that $(C/L) is a nonsingular cubic curve and $
holomorphically embeds C/L in IP . Note that
$(0) = [0,0,1], Hu.) = [l,e.,0], 1 < i < 3,
correspond to the ramification points (with index 2) of p. Topologically $(C/L)
can be constructed as follows: take 2 copies of P and cut each P from m to e.
and from e, to e,; then glue 2 copies along the cuts.
The quantities g., g,, and A depend only on the isomorphism class of L so
that we have a well-defined map
{isomorphism classes of complex tori) -♦ C
given by
[C/L] H 1728.g5(L)/A(L),
where [•] denotes the isomorphism class. (The reason for the 1728 is to have
integral coefficients in certain power series expansions.) This map is called the
modular function, and we have the important
Theoiem. The modular function is bijective.
For a proof of the above theorem see [L2] p. 39.
Let D be a divisor on M = C/L. We can write
D = S a.p., a. € I, p. € C/L.
Recall the complex vector space
L(D) = {f € H°(M,/): (f) > -D} U {0}.
287

Theorem F. Let p denote the Weierstrass function on C/L, and also let 0 = t(0)
denote the origin of C/L. Then dim L(m(0)) = m, and moreover,
L(2n(0)) = span {1, p, p', p^, p'p, ..., p'p"-2, p"};
L((2n+1)(0)) = span {1, p, p', p^, p'p, .... p", p'p"-^}.
Proof. For d > 1, any meromorphic function with (f) = 2d(0) is a constant
'OD
multiple of p ; any meromorphic function with (f) = (2d+l)(0) is a constant
multiple of p'p'^"^ □
From Theorem F we obtain nondegenerate elliptic curves
f = f|^^Qj|: C/L - IP"-\ degree f(C/L) = n.
For example, take n = 4. Then
f: z € C/L » [1, p(z), p'(z), p2(z)] € \p\
Exercise. Find a polynomial P(l,p,p',p ) = 0 of degree 4.
288

Bibliography
[ACGH] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of
Algebraic Curves, Springer-Verlag, New York, 1984.
[A] M. Agoston, Algebraic Topology, Dekker, New York, 1976.
[Ba] W. Barth, Lectures on K-3 and Enriques surfaces. Algebraic Geometry,
Lecture Notes in Math. 1124, 21-57, Springer-Verlag, New York, 1985.
[BH] E. Bombieri and D. Husemoller, Classification and embeddings of
surfaces. Algebraic Geometry, Proceed. Symposia Pure Math. 29,
329-420, Amer. Math. Soc, 1976.
[Be] L. Bers, Finite dimensional Tdchmilller spaces and generalizations. Bull.
Amer. Math. Soc, 5 (1981), 131-172.
[BPV] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces,
Springer-Verlag, New York, 1984.
[B] A. Beauville, Complex Algebraic Surfaces, Cambride Univ. Press,
Cambridge, 1983.
[BT] R. Bott and L, Tu, Differential Forms in Algebraic Topology, Springer-
Veriag, New York, 1980.
[BF] G. Bagnera and M, de Franchis, Le superficie algebriche, de quali
ammettono una rappresentazione parametrica mediante funzione
iperellitiche di due argomenti, Mem. Soc. Ital. delle Sci. Ill Ser. 15
(1908), 251-343.
[Bo] E. Bombieri, Canonical models of surfaces of general type, Publ. Math.
Inst. Hautes Etud. Sci. 42 (1973), 171-219.
[Br] G. Bredon, Sheaf Theory, McGraw-Hill, New York, 1967.
289

[Ca] F. Catanese, Canonical rings and "special" surfaces of general type.
Algebraic Geometry, Bowdoin 1985, Amer. Math. Soc.
[C] S.S. Chern, Complex Manifolds Without Potential Theory, Van
Nostrand, New York, 1967.
[CK] W.L. Chow and K. Kodaira, On analytic surfaces with two independent
meromorphic functions, Proc. Natl. Acad. Sci., 38 (1952), 319-325.
[E] F. Enriques, Le Superficie Algebraiche, Zanichelli, Bologna, 1949.
[F] W. Fulton, Algebraic Curves, Benjamin, New York, 1969.
[FK] H. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, New York,
1980.
[Fr] M. Freedman, The topology of four-dimensional manifolds, J. Diff.
Geom., 17 (1982), 357-454.
[FM] R. Friedman and J.W. Morgan, Algebraic surfaces and 4-manifolds:
Some conjectures and speculations. Bull. Amer. Math. Soc, 18 (1988),
1-20.
[G] M. Greenberg, Lectures on Algebraic Topology, Benjamin, New York,
1966.
[Gi] D. Gieseker, Global moduli for surfaces of general type. Invent. Math.,
43 (1979), 233-282.
[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley,
New York, 1978.
[Gul] R.C. Gunning, Lectures on Riemann Surfaces, Princeton Univ Press,
Princeton, 1967.
290

[Gu2] R.C. Gunning, Riemann surfaces and their associated Wirtinger
varieties. Bull. Amer. Math. Soc, 11 (1984), 287-316.
[GR] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex
Variables, Prentice-Hall, New York, 1966.
[Ha] J. Harris, Curves and their Moduli, Algebraic Geometry, Bowdoin 1989,
Amer, Math. Soc.
[Ha2] J. Harris, On varieties of minimal degree. Algebraic Geometry, Bowdoin
1985, Amer. Math. Soc.
[H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.
[H2] R. Hartshorne, On the classification of algebraic space curves, II,
Algebraic Geometry, Bowdoin 1985, Amer. Math. Soc.
[Hi] H. Hironaka, Resolution of singularities of an algebraic variety oyer a
field of characteristic zero I, II, Ann. Math., 79 (1964), 109-326.
[Hu] B. Hunt, Complex manifold geography in dimension 2 and 3, J. Diff.
Geom., 30 (1989), 51-153.
[HM] D. HusemoUer and J. Milnor, Symmetric Bilinear Forms^ Springer-
Veriag, New York, 1973.
[Hirl] F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-
Veriag, New York, 1966.
[Hir2] F. Hirzebruch, Hilbert modular surfaces. Ensign. Math., 19 (1973),
183-281.
[I] S. litaka. Algebraic Geometry, Springer-Verlag, New York, 1982.
[K] K. Kendig, Elementary Algebraic Geometry, Springer-Verlag, New York,
1974.
291

[Kn] K. Knopp, Theory of Functions II, Dover, New York, 1947.
[Kr] I. Kra, Canonical mapppings between Teichmiiller spaces. Bull. Amer.
Math. Soc, 4 (1981), 143-180.
[Kol] K. Kodaira, On Kaehler varieties of restricted type, Ann. Math., 60
(1954), 28-48.
[Ko2] K. Kodaira, On compact complex analytic surfacesi, Ann. Math., 71
(1960), 111-152. II, Ann. Math., 77 (1963), 563-626. Ill, Ann. Math.,
78 (1963), 1-40.
[Ko3] K. Kodaira, On the structure of compact analytic surfaces I, Amer. J.
Math., 86 (1964), 751-798. II, Amer. J. Math., 88 (1966), 682-721.
Ill, Amer. J. Math., 90 (1969), 55-83. IV, ibid., 1048-1066.
[KS] K. Kodaira and D. C. Spencer, Groups of complex line bundles over
compact Kaehler varieties; Divisor class groups on algebraic varieties.
Proceed. Nat. Acad. Sci. 39 (1953), 868-877.
[Ko] J. KoUar, The structure of algebraic threefolds: an introduction to
Mori's program. Bull. AMS. 17 (1987), 211-274.
[L] S. Lang, Algebra, 2nd edition, Addison-Wesley, Reading, 1984.
[L2] S. Lang, Elliptic Functions, Springer-Verlag, New York, 1987.
[M] W. Massey, Singular Homology Theory, Springer-Verlag, New York,
1980.
[Ma] H. Martens, A new proof of Torelli's theorem, Ann. Math., 78 (1963),
107-111.
[Mi] Y. Miyaoka, On the Chern numbers of general type. Invent. Math., 42
(1977), 225-237.
292

[Mul] D. Mumford, Curves and Their Jacobians, Univ. Michigan Press, Ann
Arbor, 1975.
[Mu2] D. Mumford, Tata Lectures on Theta, Birkh&user, Boston, 1983.
[N] M. Namba, Geometry of Projective Algebraic Curves, Dekker, New
York, 1984.
[Na] R. Narasimhan, Introduction to the Theory of Analytic Spaces, Lecture
Notes in Math. 25, Springer-Verlag, New York, 1966.
[00] G. Orzech and M. Orzech, Plane Algebraic Curves, Dekker, New York,
1974
[P] U. Persson, On Chern invariants of surfaces of general type. Compos.
Math., 43 (1981), 3-58.
[P2] U. Persson, An introduction to the geography of surfaces of general
type. Algebraic Geometry, Bowdoin 1985, Amer. Math. Soc.
[£c] M. Schneider, Vector bundles and submanifolds of projective space: Nine
open problems. Algebraic Geometry, Bowdoin 1985, Amer. Math. Soc.
[S] I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New
York, 1974.
[Se] J.P. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst.
Fourier, 6 (1956), 1-42.
[ST] I. Singer and J. Thorpe, Lecture Notes on Elementary Topology and
Geometry^ Springer-Verlag, New York, 1967.
[Si] Y.T. Siu, Every K-3 surface is Kaehler, Invent. Math., 73 (1983),
139-150.
293

[So] A. Sommese, Hyperplane sections of projective surfaces I - The
adjunction mapping, Duke Math. J., 46 (1979), 377-401.
[St] J. Stillwell, Classical Topology and Combinatorial Group Theory,
Springer-Verlag, New York, 1980.
[T] B. Tennison, Sheaf Theory, Cambridge Univ. Press, Cambridge, 1975.
[Ta] H-8. Tai, On Frenet frames of complex submanifolds in complex
projective spaces, Duke Math. J., 51 (1984), 163-183.
[Wal] R. Walker, Algebraic Curves, Dover, New York, 1962.
[Wa] F. Warner, Foundations of Differentiable Manifolds and Lie Groups,
Scott, Foresman and Company, Glenview, 1971.
[W] R. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall,
Englewood Cliffs, 1973.
[Wh] J.H.C. Whitehead, On simply connected 4-dimensional polyhedra.
Comment. Math. Helv., 22 (1949), 48-92.
[Ya] J. Yang, On quintic surfaces of general type. Trans. Amer. Math. Soc,
295 (1986), 431-474.
[Y] K. Yang, The Chern numbers of projective algebraic hypersurfaces.
Proceedings of the AMS Summer Research Institute, 1989.
[Y2] K. Yang, Horizontal holomorphic curves in Sp(n)-flag manifolds.
Proceed. Amer. Math. Soc., 103 (1988), 265-273.
[Y3] K. Yang, Almost Complex Homogeneous Spaces and Their Submanifolds,
World Scientific Publishing, Teaneck, 1987.
[Yal] S.T. Yau, Calabi's conjecture and some new results in algebraic
geometry, Proc. Nat. Acad. Sci., 74 (1977), 1789-1799.
294

[Ya2] S.T. Yau, On the Ricci curvature of a complex Kaehler manifold and
the complex Monge-Ampere equation. Comment. Pure and Appl.
Math., 31 (1978), 339-411.
[Z] 0. Zariski, Algebraic Surfaces, 2nd ed., Springer-Verlag, New York,
1971.
295

Page 297
Index
Abstract algebraic variety j4, j^
Abstract coordinate ring j^
Adjunction formula 69
Affine algebraic variety j_
Affine part of a projective variety 3
Algebraic curve 93
Algebraic dimension 164
Algebraic surface 154, 157
Ampleness of the canonical divisor 175
Analytic hypersurface IJ^, \J_, 1^
Analytic subvariety 79
Analytic type form 233
Analytic type function 236
Ascending chain condition 4
Associated curves of a projective curve 102, 235
B
Bagnera-de Franchis theorem 191
Base-point-free linear systems 73
Betti numbers 29, 80, 159
Bezout theorem 25

Birational classification j4, 151
Birational equivalence j4, 151
Birational map 165
Blow-up 166, 168
Boundary map 28
Canonical linear system and divisors 121
Canonical map 147
Cartan matrix 217
Catelnuovo-Enriques criterion 169
Castelnuovo theorem 185
Cech cochain 178
Chain group 29
Chern class 53, 6]_
Chern connection 228
Chow's theorem ^8, 87, 164, 236
Class 104
Closed form 32
Closure map 4
Coboundary and coboundary map 3]_
Cochain group 3]_
Cocycles 66
Coherent orientations 27
Coframe 82

Complete linear system 174, 208
Complexified cotangent bundle 48
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page 298
Complex manifold
algebraic/projective j^
compact j^
submanifolds j_^
vector bundle 5]_
Conies 93
Connection 55
matrix 55
metric 57
Coordinate ring j_3, 96
Curvature form and matrix 56, 59
Curve
algebraic 236
exceptional of the first kind 216
Cusp (simple) 100
Cycles 28
D
Dehomogenization of a polynomial 3
De Rham cohomology groups 32
Descending chain condition 4
Directed set 42
Direct image sheaf 214
Direct system 42

Divisor 64, 68
canonical 113
integral or effective 112
linear equivalence 113
principal 65, 113
Dolbeault cohomology groups 50, 75
Dolbeault isomorphism theorem 50
Dual bundle 54
Dual curve 102
E
Enriques theorem 183
Euler (topological) characteristic 158
Exact form 32
Extension of line bundles 179
Fixed part of a linear system 174
Flex (ordinary) 100
Frame 55
Frenet 235, 242
unitary (metric) 58, 82
holomorphic 58, S]_, 228
Fubini-Study metric 90, 162, 227
Fulton-Hansen theorem 251

Gap (Weierstrass) value 134
Gauss-Bonnet theorem 200
Gauss map 102
Generic subset 24
Genus formula 157
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page 299
Germs of holomorphic functions j^, 52, 104
Grassmannian manifolds 221
Grauert theorem 216
Green's operator 39
H
Hahn-Banach theorem 37
Harmonic form 36
Harmonic type 77
Hermitian structure 57
Hermitian vector bundle 57
Hessian matrix 100
Hilbert analytic NuUstellensatz 154
Hilbert basis theorem 9
Hilbert NuUstellensatz U, 17
Hironaka desingularization theorem 166
Hirzebruch-Jung strings 217
Hodge decomposition theorem 80
Hodge star operator 35
Hodge theorm for Hermitian holomorphic vector bundles 82
Holomorphic frame 58, 228
Holomorphic implicit mapping theorem j^
Holomorphic section 58
Homogeneous affine variety 2

Homogeneous coordinates 2
Homogeneous ideal 8
Homogeneous representative 101
Homogeneous set 2
Homogenization of a polynomial 3
Homology groups (simplicial) 28, 29
Hyperelliptic curve 140
Hyperplane at infinity 2
Hypersurface 21
analytic 15, 17, 19
Inhomogenous coordinates 236
Integral (effective) divisor 112
Integrality theorem 224
Intersection number 25, j^, 98
Irreducible
affine variety 96
algebraic (affine or projective) variety j^
analytic variety j//, 64
curve 184
ideal 10
J
Jacobian
inversion theorem 131

map 130
variety of a curve 126
Jordan-Holder theorem 21
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page 300
K
Kahler form 60, 162, 224, 229
Kahler manifold 78
Killing-Cartan form 227
Kodaira-Chow theorem 211
Kodaira dimension 173, 175, 211
Kodaira embedding theorem 74, 160, 227
Kodaira-Serre duality 161, 200
Kodaira-Spencer theorem 70
Kodaira vanishing theorem 83
Laplace-Beltrami operator 35, 236
Lattice (distributive) 5, 6
Lefshetz theorem on (1,1) classes 204
Leray theorem 46
Levi's extension theorem 165
Linear system 72
Linear system of plane curves 94
Line bundle 63, 66
negative and positive 83, 156
Local defining functions 64
Luroth problem 185
M

Maurer-Cartan form 162, 226
Meromorphic function 163
Miyaoka-Yau inequality 211
Minimal degree subvarieties 26
Minimal model theorem 186
Mittag-Leffler problem 47
Model 15
nonsigular 100
Moduli 180, 203
N
Nerve of an abstract simplicial complex 44
Noether-Enriques theorem 176, 187, 188
Noetherian ring 9
Noether's formula 159
Nondegenerate map 101
Normal bundle 69
Normal form 252
O
Ordinary m-ple point 105
Osculating map and sequence 236, 239, 240, 241
Osculating metric (singular) 236
Partially ordered set 4
Partition of unity 46

Period map and matrix 123, 126
Picard group 113, 181
Picard lattice 204
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page 301
Picard number 204
Plucker embedding 229, 234
Plucker formulae for plane curves 106
for space curves 237
Poincare duality 30, 152, 156, 200
dual 34, 48, 69, 85, 249
Polar class 261
Presheaf 44
Prime chain 20
Principal complex curvatures of a complex submanifold 253
Projective completion of an affine variety 2
Projective variety 2, 8
R
Rational function 163
Rational normal curve 120
Rational surface 165
Resultant of two polynomials 97
Riemann conditions 127
Riemann-Hurwitz formula 110
Riemann-Roch theorem 117, 158, 144
Ringed spaces 214
Schubert cycles 221, 223

Section 43
holomorphic 58, 63, 67
Segre embedding 170
Sheaf 42, 51
direct image 214
fine 47
germs of holomorphic sections 157
Siegel half space 128
Simple cusp 105
Simplical complex 26
Smooth point (nonsingular point) j_^
Spectrum (prime ideals) 214
Steifel manifold 226
Stein factorization theorem 197
Stokes theorem 33
Surface
elliptic 198
Enriques 197, 210
Hirzebruch 183
hyperelliptic 190
K-3 (exceptional) 193, 199
Kummer 208
minimal 169, 192
rational 183
ruled (geometrically) 169, 176

Tangent number 104
Topologically singular point 93
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation

Page 302
Torelli theorem for curves 128
Torelli theorem for K-3 surfaces 202, 206
Torus 126
Totally complex umbilic submanifold 256
Transcendence degree 20
Triangulation 27
U
Unirational projective varity 185
Universal coefficient theorem 30
Variety
abstract j4, j^
affine algebraic j_
Albanese 186
analytic j^, 48
projective algebraic 2, 8
Vector bundle 5]_
holomorphic 58
Veronese embedding 86, 95, 248
Veronese surface 26, 262
W
Weak solution 36
Weierstrass point on a Riemann surface 134

Weierstrass polynomial j^
Weierstrass preparation theorem j^
Weyl formulae (generalized) 245
Whitney product formula for Chern classes 62
Wirtinger's theorem and the degree 79, 162
Zariski topology 4
Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP] 1991 [/DP]End of Citation