Complex Geometry: An Introduction

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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