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