代数几何中的解析方法 英文版PDF电子书下载

Introduction 1

Chapter 1． Preliminary Material:Cohomology,Currents 5

1.A. Dolbeault Cohomology and Sheaf Cohomology 5

1.B. Plurisubharmonic Functions 6

1.C. Positive Currents 9

Chapter 2．Lelong numbers and Intersection Theory 15

2.A. Multiplication of Currents and Monge-Ampère Operators 15

2.B. Lelong Numbers 18

Chapter 3．Hermitian Vector Bundles, Connections and Curvature. 25

Chapter 4．Bochner Technique and Vanishing Theorems 31

4.A. Laplace-Beltrami Operators and Hodge Theory 31

4.B. Serre Duality Theorem 32

4.C. Bochner-Kodaira-Nakano Identity on K？hler Manifolds 33

4.D. Vanishing Theorems 34

Chapter 5．L2 Estimates and Existence Theorems 37

5.A. Basic L2 Existence Theorems 37

5.B. Multiplier Ideal Sheaves and Nadel Vanishing Theorem 39

Chapter 6．Numerically Effective and Pseudo-effective Line Bundles 47

6.A. Pseudo-effective Line Bundles and Metrics with Minimal Singularities 47

6.B. Nef Line Bundles 49

6.C. Description of the Positive Cones 51

6.D. The Kawamata-Viehweg Vanishing Theorem 56

6.E. A Uniform Global Generation Property due to Y.T. Siu 58

Chapter 7．A Simple Algebraic Approach to Fujita's Conjecture 61

Chapter 8．Holomorphic Morse Inequalities 71

8.A. General Analytic Statement on Compact Complex Manifolds 71

8.B. Algebraic Counterparts of the Holomorphic Morse Inequalities 72

8.C. Asymptotic Cohomology Groups 74

8.D. Transcendental Asymptotic Cohomology Functions 78

Chapter 9．Effective Version of Matsusaka's Big Theorem 83

Chapter 10．Positivity Concepts for Vector Bundles 89

Chapter 11．Skoda's L2 Estimates for Surjective Bundle Morphisms 99

11.A. Surjectivity and Division Theorems 99

11.B. Applications to Local Algebra: the Brian？on-Skoda Theorem 105

Chapter 12．The Ohsawa-Takegoshi L2 Extension Theorem 111

12.A. The Basic a Priori Inequality 111

12.B. Abstract L2 Existence Theorem for Solutions of ？-Equations 112

12.C. The L2 Extension Theorem 114

12.D. Skoda's Division Theorem for Ideals of Holomorphic Functions 122

Chapter 13．Approximation of Closed Positive Currents by Analytic Cycles 127

13.A. Approximation of Plurisubharmonic Functions Via Bergman Kernels 127

13.B. Global Approximation of Closed(1,1)-currents on a Compact Complex Manifold 129

13.C. Global Approximation by Divisors 136

13.D. Singularity Exponents and log Canonical Thresholds 143

13.E. Hodge Conjecture and approximation of(p,p)- currents 148

Chapter 14．Subadditivity of Multiplier Ideals and Fujita's Approximate Zariski Decomposition 153

Chapter 15．Hard Lefschetz Theorem with Multiplier Ideal Sheaves 159

15.A. A Bundle Valued Hard Lefschetz Theorem 159

15.B. Equisingular Approximations of Quasi Plurisubharmonic Functions 160

15.C. A Bochner Type Inequality 166

15.D. Proof of Theorem 15.1 168

15.E. A Counterexample 170

Chapter 16．Invariance of Plurigenera of Projective Varieties 173

Chapter 17．Numerical Characterization of the K？hler Cone 177

17.A. Positive Classes in Intermediate(p,p)-bidegrees 177

17.B. Numerically Positive Classes of Type(1,1) 178

17.C. Deformations of Compact K？hler Manifolds 184

Chapter 18．Structure of the Pseudo-effective Cone and Mobile Intersection Theory 189

18.A. Classes of Mobile Curves and of Mobile(n-1,n-1)-currents 189

18.B. Zariski Decomposition and Mobile Intersections 192

18.C. The Orthogonality Estimate 199

18.D. Dual of the Pseudo-effective Cone 202

18.E. A Volume Formula for Algebraic(1,1)-classes on Projective Surfaces 205

Chapter 19．Super-canonical Metrics and Abundance 209

19.A. Construction of Super-canonical Metrics 209

19.B. Invariance of Plurigenera and Positivity of Curvature of Super-canonical Metrics 216

19.C. Tsuji's Strategy for Studying Abundance 217

Chapter 20．Siu's Analytic Approach and P？un's Non Vanishing Theorem 219

References 223