Complex Geometry: An Introduction

Cover
Complex Geometry
An Introduction
ISBN 3540212906
Preface
Contents
1 Local Theory
1.1 Holomorphic Functions of Several Variables
Proposition 1.1.2
Lemma 1.1.3
Proposition 1.1.4 (Hartogs' theorem)
Definition 1.1.5
Proposition 1 . 1.6 (Weierstrass preparation theorem)
Proposition 1.1. 7 (Riemann extension theorem)
Definition 1 . 1.8
Definition 1.1.9
Proposition 1 . 1.10 (Inverse function theorem)
Proposition 1.1.11 (Implicit function theorem)
Corollary 1.1.12
Proposition 1. 1.13
Definition 1.1.14
Proposition 1.1.15
Definition 1.1 .16
Proposition 1.1.17 (Weierstrass division theorem)
Proposition 1. 1.18
Corollary 1.1.19
Remark 1.1.20
Definition 1 . 1.21
Definition 1.1.22
Definition 1 .1.23
Definition 1.1 .24
Lemma 1.1 .25
Lemma 1.1 .26
Definition 1.1.27
Lemma 1.1.28
Proposition 1.1.29 (Nullstellensatz)
Theorem 1.1 .30
Definition 1.1.31
Definition 1.1 .33
Definition 1.1.34
Proposition 1.1.35
Proposition 1.1 .36 (Schwarz lemma)
Exercises
1.2 Complex and Hermitian Structures
Definition 1.2.1
Lemma 1.2.2
Corollary 1.2.3
Definition 1.2.4
Lemma 1.2.5
Lemma 1.2.6
Definition 1.2. 7
Proposition 1.2.8
Lemma 1.2.9
Definition 1.2.10
Definition 1 .2.11
Definition 1.2.13
Lemma 1.2.14
Lemma 1.2.15
Lemma 1.2.16
Lemma 1.2.17
Definition 1.2.18
Proposition 1 .2.20
Definition 1.2.21
Remark 1.2.22
Lemma 1.2.23
Lemma 1 .2.24
Definition 1.2.25
Proposition 1 .2.26
Corollary 1 .2.27
Corollary 1.2.28
Definition 1.2.29
Proposition 1.2.30
Proposition 1.2.31
Definition 1.2.35
Corollary 1.2.36 (Hodge-Riemann bilinear relation)
Exercises
1.3 Differential Forms
Proposition 1.3.1
Proposition 1.3.2
Definition 1.3.3
Corollary 1.3.4
Definition 1.3.5
Lemma 1 .3.6
Proposition 1.3.7 (a -Poincare lemma in one variable)
Proposition 1.3.8 ( 8-Poincare lemma in several variables)
Corollary 1.3.9 (8 -Poincare lemma on the open disc)
Definition 1.3.11
Proposition 1.3.12
Exercises
2 Complex Manifolds
2.1 Complex Manifolds: Definition and Examples
Definition 2.1.1
Definition 2.1.2
Definition 2.1.3
Definition 2.1.4
Proposition 2.1.5
Corollary 2.1.6
Definition 2.1.7
Definition 2.1.8
Proposition 2.1.9 (Siegel)
Definition 2.1.10
Affine space
Projective space
Complex tori
Affine hypersurfaces
Projective hypersurfaces
Complete intersections
Complex Lie groups
Quotients
Definition 2.1.11
Proposition 2.1.13
Ball quotients
Finite quotients of product of curves
Hopf manifolds
lwasawa manifold
Grassmannian manifolds
Flag manifolds.
Definition 2.1.16
Definition 2.1.17
Definition 2.1.18
Exercises
Comments
2.2 Holomorphic Vector Bundles
Definition 2.2.1
Remarks 2.2.2
Meta-theorem 2.2.3
Definition 2.2.5
Proposition 2.2.6
Definition 2.2.7
Proposition 2.2.9
Corollary 2.2.10
Corollary 2.2.11
Definition 2.2.12
Definition 2.2.13
Definition 2.2.14
Lemma 2.2.15
Definition 2.2.16
Proposition 2.2.17 (Adjunction formula)
Definition 2.2.18
Proposition 2.2.19
Definition 2.2.21
Definition 2.2.23
Lemma 2.2.24
Definition 2.2.25
Definition 2.2.26
Definition 2.2.27
Proposition 2.2.28
Exercises
Comments
2.3 Divisors and Line Bundles
Definition 2.3.1
Definition 2.3.2
Definition 2.3.4
Definition 2.3.5
Definition 2.3.7
Definition 2.3.8
Proposition 2.3.9
Corollary 2.3.10
Proposition 2.3.12
Corollary 2.3.13
Lemma 2.3.14
Remark 2.3.15
Definition 2.3.16
Remarks 2. 3. 17
Proposition 2.3.18
Corollary 2.3.19
Corollary 2.3.20
Remark 2.3.21
Lemma 2.3.22
Definition 2.3.23
Definition 2.3.25
Proposition 2.3.26
Definition 2.3.28
Veronese map.
Segre map.
Definition 2.3.29
Proposition 2.3.30
Lemma 2.3.31
Corollary 2.3.32
Definition 2.3.33
Proposition 2.3.34
Exercises
Comments
2.4 The Projective Space
Proposition 2.4.1
Corollary 2.4.2
Proposition 2.4.3
Proposition 2.4.4 (Euler sequence)
Corollary 2.4.6
Proposition 2.4. 7
Corollary 2.4.9
Exercises
2.5 Blow-ups
Example 2. 5. 1 Blow-up of a point
Example 2.5.2 Blow-up along a linear subspace
Proposition 2.5.3
Definition 2.5.4
Proposition 2.5.5
Corollary 2.5.6
Definition 2.5.
Proposition 2.5.8
Remark 2.5.9
Exercises
Comments
2.6 Differential Calculus on Complex Manifolds
Definition 2.6.1
Proposition 2.6.2
Proposition 2.6.4
Definition 2.6.5
Definition 2.6. 7
Corollary 2.6.8
Definition 2.6.9
Proposition 2.6.10
Proposition 2.6. 11
Corollary 2.6.12
Definition 2.6.13
Corollary 2.6.14
Proposition 2.6.15
Definition 2.6.16
Proposition 2.6.17
Corollary 2.6.18
Theorem 2.6.19 (Newlander-Nierenberg)
Definition 2.6.20
Corollary 2.6.21
Definition 2.6.22
Lemma 2.6.23
Definition 2.6.24
Corollary 2.6.25
Theorem 2.6.26
Exercises
Comments
3 Kahler Manifolds
3.1 Kahler Identities
Definition 3.1.1
Corollary 3.1.2
Definition 3.1.3
Lemma 3.1.4
Definition 3.1.5
Definition 3.1.6
Lemma 3.1.7
Corollary 3.1.8
Proposition 3.1.10
Corollary 3.1.11
Proposition 3.1.12 (Kahler identities)
Definition 3.1.13
Remark 3.1.14
Exercises
Comments
3.2 Hodge Theory on Kahler Manifolds
Definition 3.2.1
Proposition 3.2.2
Lemma 3.2.3
Definition 3.2.4
Lemma 3.2.5
Proposition 3.2.6
Remarks 3. 2. 7
Theorem 3.2.8 (Hodge decomposition)
Corollary 3. 2. 9
Corollary 3.2.10 (80-lemma)
Remark 3.2.11
Corollary 3.2.12
Remark 3.2.13
Definition 3.2.14
Exercises
Comments
3.3 Lefschetz Theorems
Lemma 3.3. 1
Proposition 3.3.2 (Lefschetz theorem on (1, 1)-classes)
Remark 3.3.3
Definition 3.3.4
Definition 3.3.5
Corollary 3.3.6
Definition 3.3. 7
Proposition 3.3.8
Examples 3.3.9
Corollary 3.3.10
Definition 3.3.11
Remark 3.3.12
Proposition 3.3.13 (Hard Lefschetz theorem)
Corollary 3.3.14
Proposition 3.3.15 (Hodge-Riemann bilinear relation)
Corollary 3.3.16 (Hodge index theorem)
Remark 3. 3.17
Corollary 3.3.18
Definition 3.3.19
Definition 3.3.20
Hodge Conjecture
Remarks 3.3.21
Exercises
Comments
Appendix to Chapter 3
3.A Formality of Compact Kahler Manifolds
Definition 3.A.l
Definition 3.A.2
Lemma 3.A.3
Definition 3.A.4
Lemma 3.A.5
Definition 3.A.6
Definition 3.A.8
Definition 3.A.10
Lemma 3.A.12
Remark 3.A.13
Examples 3.A.14
Definition 3.A.15
Proposition 3.A.16
Examples 3.A.17
Definition 3.A.18
Remark 3.A.19
Definition 3.A.20
Definition 3.A.21
Lemma 3.A.22 (dd^c-lemma)
Remark 3.A.23
Corollary 3.A.24
Definition 3.A.25
Corollary 3.A.26
Corollary 3.A.27
Proposition 3.A.28
Remark 3.A.29
Remark 3.A.30
Definition 3.A.31
Remark 3.A.32
Proposition 3.A.33
Example 3.A.34
Remark 3.A.35
Exercises
3.B SUSY for Kahler Manifolds
Definition 3.B.l
Example 3.B.2
Definition 3.B.3
Riemannian geometry
Complex geometry
Kahler geometry
Hermitian geometry
Beyond.
3.C Hodge Structures
Definition 3.C.l
Proposition 3.C.2
Examples 3. C. 3
Definition 3.C.4
Definition 3.C.5
Lemma 3.C.6
Example 3. C. 7
Remark 3. C.S
Example 3. C. 9
Proposition 3.C.10
Proposition 3.C.ll
Comments
4 Vector Bundles
4.1 Hermitian Vector Bundles and Serre Duality
Definition 4.1.1
Examples 4.1.2
Example 4.1.3
Proposition 4.1.4
Example 4.1.5
Definition 4.1.6
Definition 4.1. 7
Definition 4.1.9
Definition 4.1.10
Definition 4.1.11
Lemma 4.1.12
Theorem 4.1.13 (Hodge decomposition)
Corollary 4.1.14
Proposition 4.1.15 (Serre duality)
Corollary 4.1.16
Remark 4.1.11
Exercises
Comments
4.2 Connections
Definition 4.2.1
Definition 4.2.2
Proposition 4.2.3
Corollary 4.2.4
Remark 4.2.5
Examples 4.2.6
Lemma 4.2. 7
Definition 4.2.8
Definition 4.2.9
Definition 4.2.10
Corollary 4.2.11
Definition 4.2.12
Corollary 4.2.13
Proposition 4.2.14
Example 4.2.15
Examples 4.2.16
Definition 4.2.17
Definition 4.2.18
Proposition 4.2.19
Remarks 4.2.20
Exercises
Comments
4.3 Curvature
Definition 4.3.1
Lemma 4.3.2
Examples 4.3.3
Lemma 4.3.4
Lemma 4.3.5 (Bianchi identity)
Example 4.3.6
Proposition 4.3. 7
Proposition 4.3.8
Proposition 4.3.10
Remark 4.3.11
Example 4.3.12
Definition 4.3.14
Definition 4.3.15
Remark 4.3.16
Lemma 4.3.17
Proposition 4.3.18
Corollary 4.3.19
Example 4.3.20
Remark 4.3.21
Exercises
Comments
4.4 Chern Classes
Definition 4.4.1
Lemma 4.4.2
Proposition 4.4.3
Lemma 4.4.4
Corollary 4.4.5
Lemma 4.4.6
Remark 4.4. 7
Examples 4.4.8
Remarks 4.4.9
Definition 4.4.10
Example 4.4.11
Proposition 4.4.12
Proposition 4.4.13
Exercises
Comments
Appendix to Chapter 4
4.A Levi-Civita Connection and Holonomy on Complex Manifolds
Definition 4.A.1
Lemma 4.A.2
Theorem 4.A.3
Proposition 4.A.4
Corollary 4.A.5
Lemma 4.A.6
Proposition 4.A. 7
Proposition 4.A.8
Proposition 4.A.9
Definition 4.A.10
Proposition 4.A.ll
Lemma 4.A.12
Definition 4.A.13
Proposition 4.A.14
Theorem 4.A.15 (de Rham)
Theorem 4.A.16 (Berger)
Holonomy principle.
Proposition 4.A.17
Proposition 4.A.18
Exercises
4.B Hermite-Einstein and Kahler-Einstein Metrics
Definition 4.B.l
Example 4.B.2
Definition 4.B.3
Lemma 4.B.4
Remark 4.B.S
Proposition 4.B.6
Remarks 4.B. 7
Definition 4.B.8
Theorem 4.B.9 (Donaldson, Uhlenbeck, Yau)
Definition 4.B.ll
Definition 4.B.12
Corollary 4.B.13
Corollary 4.B.14
Remark 4.B. 15
Examples 4.B.l6
Proposition 4.B.17
Remark 4.B.l8
Theorem 4.B.19 (Calabi-Yau)
Lemma 4.B.20
Proposition 4.B.21
Corollary 4.B.22
Corollary 4.B.23
Theorem 4.B.24 (Aubin, Yau)
Exercises
5 Applications of Cohomology
5.1 Hirzebruch-Riemann-Roch Theorem
Theorem 5.1.1 (Hirzebruch-Riemann-Roch)
Examples 5.1.2
Definition 5.1.3
Corollary 5.1.4
Grothendieck-Riemann-Roch formula
Atiyah-Singer Index Theorem
Remark 5.1.
Exercises
Comments
5.2 Kodaira Vanishing Theorem and Applications
Definition 5.2.1
Proposition 5.2.2 (Kodaira vanishing)
Lemma 5.2.3 (Nakano identity)
Lemma 5.2.4
Example 5.2. 5
Proposition 5.2.6 (Weak Lefschetz theorem)
Proposition 5.2. 7
Corollary 5.2.8 (Grothendieck lemma)
Exercises
Comments
5.3 Kodaira Embedding Theorem
Proposition 5.3.1 (Kodaira embedding theorem)
Lemma 5.3.2
Corollary 5.3.3
Definition 5.3.4
Corollary 5.3.
Exercises
Comments
6 Deformations of Complex Structures
6.1 The Maurer-Cartan Equation
Remark 6.1.1
Lemma 6.1.2
Definition 6.1.3
Lemma 6.1.4
Proposition 6.1.5
Corollary 6.1.6
Definition 6.1.
Lemma 6.1.8
Lemma 6.1.9 (Tian-Todorov lemma)
Corollary 6.1.10
Proposition 6.1.11
Exercises
6.2 General Results
Examples 6.2.1
Proposition 6.2.2 (Ehresmann)
Corollary 6.2.3
Theorem 6.2.4
Theorem 6.2.5 (Kodaira
Definition 6.2.6
Definition 6.2.8
Proposition 6.2.10
Definition 6.2.11
Theorem 6.2.12 (Kodaira-Spencer)
Theorem 6.2.13 (Kuranishi)
Comments
Appendix to Chapter 6
6.A dGBV-Algebras
Definition 6.A.l
Proposition 6.A.2
Proposition 6.A.3
Lemma 6.A.4
Corollary 6.A.5
Remark 6.A.6
Proposition 6.A.7 (Generalized Tian-Todorov lemma)
Corollary 6.A.8
Definition 6.A.10
Proposition 6.A.11
Proposition 6.A.12
Comments:
A Hodge Theory on Differentiable Manifolds
Definition A.O.l
Definition A.0.2
Proposition A.0.3 (Poincare lemma)
Definition A.0.4
Definition A.0.5
Proposition A.0.6 (Poincare duality)
Examples A. 0. 7
Definition A.0.8
Lemma A.0.9
Definition A.O.ll
Lemma A.0.12
Corollary A.0.13
Lemma A.0.14
Corollary A.0.15 (Poincare duality)
Theorem A.0.16 (Hodge decomposition)
Corollary A.0.17
Lemma A.0.18
Comments
B Sheaf Cohomology
Definition B.0.19
Definition B.0.21
Definition B.0.23
Definition B.0.24
Definition B.0.25
Definition B.0.26
Definition B.0.27
Corollary B.0.28
Definition B.0.29
Definition B.0.30
Lemma B.0.31
Proposition B.0.32
Definition B.0.33
Proposition B.0.34
Proposition B.0.35
Definition B.0.36
Corollary B.0.37
Definition B.0.38
Proposition B .0.39
Definition B.0.40
Examples B. 0.41
Proposition B.0.42
Proposition B.0.43
Corollary B.0.44
Definition B.0.45
Definition B.0.46
Comment
References
Index