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