Complex Manifolds (AMS Chelsea Publishing)

204 Pages·2006·8.07 MB·English

by  James Morrow and Kunihiko Kodaira

COMPLEX MANIFOLDS
JAMES MORROW
KUNIHIKO I(oDAIRA
AMS CHELSEA PUBLISHING
American Mathematical Society· Providence, Rhode Island
2000 Mathematics Subject Classification. Primary 32Qxx.
Library of Congress Cataloging-in-Publication Data
Morrow, James A., 1941-
Complex manifolds / James Morrow, Kunihiko Kodaira.
p. cm.
Originally published: New York: Holt, Rinehart and Winston, 1971.
Includes bibliographical references and index.
ISBN 0-8218-4055-X (alk. paper)
1. Complex manifolds. I. Kodaira, Kunihiko, 1915- II. Title.
QA331.M82 2005
515'.946---dc22 20051
© 1971 held by the American Mathematical Society.
Reprinted with errata by the American Mathematical Society, 2006
Printed in the United States of America.
§ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10987654321 11 10 09 08 07 06
Preface
The study of algebraic curves and surfaces is very classical. Included
among the principal investigators are Riemann, Picard, Lefschetz, Enriques,
Severi, and Zariski. Beginning in the late 1940s, the study of abstract (not
necessarily algebraic) complex manifolds began to interest many mathe
maticians. The restricted class of Kahler manifolds called Hodge manifolds
turned out to be algebraic. The proof of this fact is sometimes called the
Kodaira embedding theorem, and its proof relies on the use of the vanishing
theorems for certain cohomology groups on Kahler manifolds with positive
lines fundles proved somewhat earlier by Kodaira. This theorem is analogous
to the theorem of Riemann that a compact Riemann surface is algebraic.
This book is a revision and organization of a set of notes taken from the
lectures of Kodaira at Stanford University in 1965-1966. One of the main
points was to give the original proof of the Kodaira embedding theorem.
There is a generalization of this theorem by Grauert. Its proof is not included
here.
Beginning in the mid-1950s Kodaira and Spencer began the study of
deformations of complex manifolds. A great deal of this book is devoted to
the study of deformations. Included are the semicontinuity theorems and the
local completeness theorem of Kuranishi. There has also been a great deal
accomplished on the classification of complex surfaces (complex dimension
2). That material is not included here.
The outline is roughly as follows. Chapter I includes some of the basic
ideas such as surgery, quadric transformations, infinitesimal deformations,
deformations. In Chapter 2, sheaf cohomology is defined and some of the
completeness theorems are proved by power series methods. The de Rham
and Dolbeault theorems are also proved. In Chapter 3 Kahler manifolds
are studied and the vanishing and embedding theorems are proved. In Chapter
4 the theory of elliptic partial differential equations is used to study the
semi-continuity theorems and Kuranishi's theorem.
It will help the reader if he knows some algebraic topology. Some results
from elliptic partial differential equations are used for which complete
references are given. The sheaf theory is self-contained.
We wish to thank the publisher for patience shown to the authors and
Nancy Monroe for her excellent typing.
James A. Morrow
Seattle, Washington Kunihiko Kodaira
January 1971
v

Contents
Preface v
Chapter 1. Definitions and Examples of Complex Manifolds 1
1. Holomorphic Functions 1
2. Complex Manifolds and Pseudogroup Structures 7
3. Some Examples of Construction (or Description) of
Compact Complex Manifolds 11
4. Analytic Families; Deformations 18
Chapter 2. Sheaves and Cohomology 27
1. Germs of Functions 27
2. Cohomology Groups 30
3. Infinitesimal Deformations 35
4. Exact Sequences 56
5. Vector Bundles 62
6. A Theorem of Dolbeault (A fine resolution of (I)) 73
Chapter 3. Geometry of Complex Maoifolds 83
1. Hermitian Metrics; Kahler Structures 83
2. Norms and Dual Forms 92
3. Norms for Holomorphic Vector Bundles 100
4. Applications of Results on Elliptic Operators 102
5. Covariant Differentiation on Kahler Manifolds 106
6. Curvatures on Kahler Manifolds 116
7. Vanishing Theorems 125
8. Hodge Manifolds 134
Chapter 4. Applications of Elliptic Partial Differential Equations to
Deformations 147
1. Infinitesimal Deformations 147
2. An Existence Theorem for Deformations I.
(No Obstructions) 155
3. An Existence Theorem for Deformations II. (Kuranishi's
Theorem) 165
4. Stability Theorem 173
Bibliography 186
Index 189
Errata 193
vii

Complex Manifolds

[1]
Definitions and Examples
of Complex Manifolds
I. Holomorphic Functions
The facts of this section must be well known to the reader. We review
them briefly.
DEFINITION 1.1. A complex-valued function J(z) defined on a connected
open domain W s;;; en is called hoiomorphic, if for each a = (a1> "', an) e W,
J(z) can be represented as a convergent power series
+00
L ek, ... kn(Z1 - a1)k, ... (zn - a,,)k"
k,~O.kn~O
in some neighborhood of a.
= =
REMARK. Ifp (z) LCk ... kn (Z1 - a1)k, •.• (z" - an)k" converges at z w, then
p(z) converges for any z such that IZk - akl < IWk - akl for 1 :S k :S n.
Proof We may assume a = O. Then there is a constant C> 0 such that
for all coefficients Ck .... kn'
Ie W"l .•• wknl < C
k, .. ·kn 1 ,,_.
Hence
I
I
Ie zk, ... zknl < C 2 Ik' '" 2 Ik" (1)
k, ... kn 1 ,,- Z Z •
W1 W"
If Izdwil < 1 for 1 :S i:S n, (1) gives
I)
n(
L Ie", "'knZ~' '" zktl :S C. 1 < + 00. Q.E.D.
1=1 Zi
1- -
Wi
1