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