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AMS/IP https://doi.org/10.1090/amsip/018
Studies in
Advanced
Mathematics
Volume 18
Complex Differentia l
Geometry
Fangyang Zheng
American Mathematical Society • International Press
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Shing-Tung Yau, Managing Edito r
2000 Mathematics Subject Classification. Primar y 53 01;
Secondary 53-00, 53-02, 53C55.
The author was supported i n part by NSF Grant # DMS-9703884 ,
NSA Grant MDA904-98-1-0036 , and a fellowship from the Alfred P . Sloan Foundation .
Library of Congress Cataloging-in-Pubiicatio n Dat a
Zheng, Fangyang, 1962-
Complex differential geometry / Fangyang Zheng.
p. cm. — (AMS/IP studies in advanced mathematics, ISSN 1089-3288 ; v. 18)
Includes bibliographical references and index.
ISBN 0-8218-2163-6
1. Complex manifolds. 2 . Geometry, Differential. I . Title. II . Series.
QA331.7.Z48 200 0
516.3'6—<ic21 00-05654 9
Copying an d reprinting . Individua l reader s of this publication, an d nonprofi t librarie s
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To Qun and Ale x
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Contents
Preface vii
Part 1 . Riemannia n Geometr y 1
Chapter 1. Differentiat e Manifold s and Vector Bundles 3
1.1. Differentiabl e Manifold s 3
1.2. Tangen t Spaces and Vector Fields 5
1.3. Vecto r Bundles 8
1.4. Tangen t Bundles and Tensor Fields 1 1
1.5. Th e Topology of Smooth Manifolds 1 4
1.6. Li e Groups and Lie Algebras 1 6
Appendix: Topology , Homotopy and Covering Spaces 2 0
Exercises 2 3
Chapter 2. Metric , Connection, and Curvature 2 9
2.1. Metric , Connection, and Curvature 2 9
2.2. Linea r Connections and Geodesies 3 1
2.3. Riemannia n Metrics and Riemannian Connections 3 5
2.4. Sectional , Ricci and Scalar Curvatures 3 7
2.5. Cartan' s Structure Equations and Examples 4 0
Exercises 4 4
Chapter 3. Th e Geometry of Complete Riemannian Manifolds 4 9
3.1. Riemannia n Distance 4 9
3.2. Completenes s and Hopf-Rinow Theorem 5 3
3.3. Jacob i Fields and Conjugate Points 5 6
3.4. Car t an-Ambrose-Hicks Theorem and Space Forms 6 4
3.5. Homogeneou s and Symmetric Spaces 66
3.6. Hodg e Theorem and Comparison Theorems 7 0
Exercises 7 4
Part 2 . Comple x Manifold s 8 1
Chapter 4. Comple x manifolds and Analytic Varieties 8 3
4.1. Holomorphi c Functions of One or More Complex Variables 8 3
4.2. Definitio n and Examples of Complex Manifolds 8 6
4.3. Th e Almost Complex Structure 8 9
4.4. Mor e Examples 9 2
4.5. Hypersurface s and Analytic Subvarieties 9 5
4.6. Divisor s and Analytic Cycles 9 9
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vi CONTENT S
Exercises 10 1
Chapter 5. Holomorphi c Vector Bundles, Sheaves and Cohomology 10 5
5.1. Holomorphi c Vector Bundles 10 5
5.2. Sheave s 10 8
5.3. Shea f Cohomology Groups 11 2
5.4. Holomorphi c Line Bundles 11 5
5.5. Cher n Classes 11 9
Exercises 12 3
Chapter 6. Compac t Complex Surfaces 12 7
6.1. Th e Topological Invariants 12 7
6.2. Th e Kodaira Dimension and the Algebraic Dimension 13 2
6.3. Example s of Surfaces 13 7
6.4. Enriques-Kodair a Classification Theory for Surfaces 14 2
Exercises 15 1
Part 3 . Kahle r Geometry 15 5
Chapter 7. Hermitia n and Kahler Metrics 15 7
7.1. Connection s on Vector Bundles and Their Curvature 15 7
7.2. Cher n Forms of a Complex Vector Bundle 16 0
7.3. Hermitia n Bundles 16 6
7.4. Hermitia n and Kahler Metrics on Complex Manifolds 17 0
7.5. Th e Curvature of a Hermitian or Kahler Metric 17 6
7.6. Wu' s Theorem, Schwarz Lemma and Hartogs Phenomenon 18 1
Exercises 18 7
Chapter 8. Compac t Kahler Manifolds 19 1
8.1. Hodg e Theorem and Hodge Decomposition 19 1
8.2. Th e Hard Lefschetz Theorem 19 5
8.3. Kodair a Vanishing and Embedding Theorems 20 0
8.4. Ampl e Subvarieties and Ample Vector Bundles 20 4
8.5. Hermitia n Symmetric Spaces and Kahler C-Spaces 20 8
8.6. Th e Hartshorne-Prankel Conjecture 21 3
Exercises 21 9
Chapter 9. Kahle r Geometry 22 1
9.1. Calabi' s Conjecture and Kahler-Einstein Metrics 22 1
9.2. Corollarie s of Yau's Theorems 22 5
9.3. Invarian t Metrics 23 0
9.4. Harmoni c Maps and the Rigidity Theorems 23 6
9.5. Non-positivel y Curved Kahler Surfaces 24 2
Exercises 24 9
Bibliography 25 5
Index 259
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Preface
Complex manifold ha s been sitting in the overlap of quite a few branches of
mathematics, suc h a s differential geometry , algebrai c geometry , severa l comple x
variables, global analysis, topology, algebraic number theory, mathematical physics,
etc..
On one hand, complex manifolds provide a rich class of geometric objects. Fo r
instance, th e (common ) zer o locus of any (generi c set of ) comple x polynomial s
is always a complex manifold. I n fact, unti l very recently, the connected sum of
simply connected algebraic surfaces has been the only known examples of simply
connected smoot h 4-manifolds . O n the othe r hand , comple x manifold s behav e
rather differently tha n generic smooth manifolds. The y are much more coherent.
The existence of a complex structure often imposes topological consequences on the
underlying smooth manifold. Fo r instance, only a rather special kind of compact
smooth manifold ca n admit a Kahlerian complex structure. A s another example,
the group of biholomorphisms of a compact complex manifold is finite dimensional,
while the diffeomorphism group of a smooth manifold is always infinite dimensional.
This rich yet restrictive character makes complex manifold a rather special and
interesting object o f study. Thi s book is about the differential geometri c aspect s
of complex manifolds. Ou r main goal is to provide a self-contained, comprehensiv e
approach t o th e subject , an d savin g the reader s fro m travelin g bac k an d fort h
between two sets of books in learning the subject. Also , in the last chapter, w e
include some more recent results in the field, most of which are not yet collected in
textbooks.
The book is divided into three parts, each contains three chapters. Th e firs t
part contain s standar d material s fro m genera l topology , differentiabl e manifold ,
and basic Riemannian geometry. Th e second part discuss complex manifolds an d
analytic varieties, sheaves and holomorphic vector bundles, as well as a brief account
of the surface classification theory , which provide the readers with some concrete
examples of complex manifolds. Th e last part is the main purpose of the book, in
which we discuss metric, connection, curvature and the various roles they played in
the study of complex manifolds. W e also collected some exercises at the end of each
chapter. Mos t of them are ranging from straight forward to medium challenging,
and serve the purpose of enhancing the comprehension of the topics. A few of them
are more challenging, and could be regarded as questions leading to supplementary
readings.
Vll
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Vlll PREFACE
Part of the materials in this book have been used by the author in the lectures
of a graduate course given at Ohio State University in 1998/99, as well as in a mini
course taught i n the 199 8 summer schoo l at Nanjin g University . W e would like
to thank the students who participated for their feedbacks and suggestions, which
made the book relatively more focused and readable. We would also like to thank
Eric Loubeau, G . Tian, H. Wu and S-T Yau for several valuable corrections and
suggestions towards the manuscript. Finally , we would like to thank NSF, NSA,
Sloan Foundation and the Ohio State University for the financial support , whic h
made the writing of this book possible.
Fangyang Zheng
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Part 1
Riemannian Geometr y
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