本书作者Jean-Pierre Demailly 教授是法国格勒诺布尔第一大学数学系教授，著名数学家，1994年获选为法国科学院院士。本书是全英文版，讲述了代数几何中的分析方法，该方法广泛地应用于线性系列，代数向量丛的消失定理等。

This volume is an expansion of lectures given by the author at the Park City Mathematics Institute in 2008 as well as in other places. The main purpose of the book is to describe analytic techniques which are useful to study questions such as linear series, multiplier ideals and vanishing theorems for algebraic vector bundles. The exposition tries to be as condensed as possible, assuming that the reader is already somewhat acquainted with the basic concepts pertaining to sheaf theory,homological algebra and complex differential geometry. In the final chapters, some very recent questions and open problems are addressed, for example results related to the finiteness of the canonical ring and the abundance conjecture, as well as results describing the geometric structure of Kahler varieties and their positive cones.

Introduction

Chapter 1. Preliminary Material: Cohomology, Currents

1.A. Dolbeault Cohomology and Sheaf Cohomology

1.B. Plurisuhharmonic Functions

1.C. Positive Currents

Chapter 2. Lelong numbers and Intersection Theory

2.A. Multiplication of Currents and Monge-Ampere Operators

2.B. Lelong Numbers

Chapter 3. Hermitian Vector Bundles,Connections and Curvature

Chapter 4. Bochner Technique and Vanishing Theorems

4.A. Laplace-Beltrami Operators and Hodge Theory

4.B. Serre Duality Theorem

4.CBochner-Kodaira-Nakano Identity on Kahler Manifolds

4.D. Vanishing Theorems

Chapter 5. L2 Estimates and Existence Theorems

5.A. Basic L2 Existence Theorems

5.B. Multiplier Ideal Sheaves and Nadel Vanishing Theorem

Chapter 6. Numerically Effective andPseudo-effective Line Bundles

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

6.B. Nef Line Bundles

6.C. Description of the Positive Cones

6.D. The Kawamat~-Viehweg Vanishing Theorem

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

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

Chapter 8. Holomorphic Morse Inequalities

8.A. General Analytic Statement on Compact Complex Manifolds

8.B. Algebraic Counterparts of the Holomorphic Morse Inequalities

8.C. Asymptotic Cohomology Groups

8.D. Transcendental Asymptotic Cohomology Functions

Chapter 9. Effective Version of Matsusaka's Big Theorem

Chapter 10. Positivity Concepts for Vector Bundles

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

11.A. Surjectivity and Division Theorems

11.B. Applications to Local Algebra: the Brianqon-Skoda Theorem

Chapter 12. The Ohsawa-Takegoshi L2 Extension Theorem

12.A. The Basic a Priori Inequality

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

12.C. The L2 Extension Theorem

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

Chapter 13. Approximation of Closed Positive Currents

by Analytic Cycles

13.A. Approximation of Plurisubharmonic Functions Via Bergman kernels

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

13.C. Global Approximation by Divisors

13.D. Singularity Exponents and log Canonical Thresholds

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

Chapter 14. Subadditivity of Multiplier Ideals

and Fujita's Approximate Zariski Decomposition

Chapter 15. Hard Lefschetz Theorem

with Multiplier Ideal Sheaves

15.A. A Bundle Valued Hard Lefschetz Theorem

15.B. Equisingular Approximations of Quasi Plurisubharmonic Functions

15.C. A Bochner Type Inequality

15.D. Proof of Theorem 15.1

15.E. A Counterexample

Chapter 16. Invariance of Plurigenera of Projective Varieties

Chapter 17. Numerical Characterization of the K~ihler Cone

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

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

17.C. Deformations of Compact K~hler Manifolds

Chapter 18. Structure of the Pseudo-effective Cone

and Mobile Intersection Theory

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

18.B. Zariski Decomposition and Mobile Intersections

18.C. The Orthogonality Estimate

18.D. Dual of the Pseudo-effective Cone

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

Chapter 19. Super-canonical Metrics and Abundance

19.A. Construction of Super-canonical Metrics

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

19.C. Tsuji's Strategy for Studying Abundance

Chapter 20. Siu's Analytic Approach and Paun's

Non Vanishing Theorem

References