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.

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Introduction
Chapter 1. Preliminary Material: Cohomology, Currents
Chapter 2. Lelong numbers and Intersection Theory
Chapter 3. Hermitian Vector Bundles, Connections and Curvature
Chapter 4. Bochner Technique and Vanishing Theorems
Chapter 5. L2 Estimates and Existence Theorems
Chapter 6. Numerically E ective andPseudo-e ective Line Bundles
Chapter 7. A Simple Algebraic Approach to Fujita’s Conjecture
Chapter 8. Holomorphic Morse Inequalities
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
Chapter 12. The Ohsawa-Takegoshi L2 Extension Theorem
Chapter 13. Approximation of Closed Positive Currents by Analytic Cycles
Chapter 14. Subadditivity of Multiplier Ideals and Fujita’s Approximate Zariski Decomposition
Chapter 15. Hard Lefschetz Theorem with Multiplier Ideal Sheaves
Chapter 16. Invariance of Plurigenera of Projective Varieties
Chapter 17. Numerical Characterization of the Kahler Cone
Chapter 18. Structure of the Pseudo-e ective Cone and Mobile Intersection Theory
Chapter 19. Super-canonical Metrics and Abundance
Chapter 20. Siu’s Analytic Approach and P aun’s Non Vanishing Theorem
References

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