The main purpose of this 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. One century after the ground-breaking work of Riemann on geometric aspects of function theory， the general progress achieved in differential geometry and global analysis on manifolds resulted into major advances in the theory of algebraic and analytic varieties of arbitrary dimension. One central unifying concept is positivity， which can be viewed either in algebraic terms （positivity of divisors and algebraic cycles）， or in more analytic terms （plurisubharmonicity， Hermitian connections with positive curvature）. In this direction， one of the most basic results is Kodairas vanishing theorem for positive vector bundles （1953——1954）， which is a deep consequence of the Bochner technique and the theory of harmonic forms initiated by Hodge during the 1940s. This method quickly led Kodaira to the well-known embedding theorem for projective varieties， a far reaching extension of Riemanns characterization of abelian varieties. Further refinements of the Bochner technique led ten years later to the theory of L2 estimates for the Cauchy-Riemann operator， in the hands of Kohn， Andreotti-Vesentini and Hormander among others. Not only can vanishing theorems be proved or reproved in that manner， but perhaps more importantly， extremely precise information of a quantitative naturc can be obtained about solutions of -equations， their zeroes，poles and growth at infinity. We try to present here a condensed exposition of these techniques， assuming that the reader is already somewhat acquainted with the basic concepts pertaining to sheaf theory， cohomology and complex differential geometry. In the final chapter， we address very recent questions and open problems， e.g. 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. This book is an expansion of lectures given by the author at the Park City Mathematics Institute in 2008 and was published partly in Analytic and Algebraic Geometry， edited by Jeff McNeal and Mircea Mustata， It is a volume in the Park City Mathematics Series， a co-publication of the Park City Mathematics Institute and the American Mathematical Society.
This volume is an expaion of lecturesgiven by the author at the Park City Mathematics Ititute in 2008 aswell as in other places. The main purpose of the book is todescribe analytic techniques which are useful to study questio suchas linear series, multiplier ideals and vanishing theorems foralgebraic vector bundles. The exposition tries to be as condeed aspossible, assuming that the reader is already somewhat acquaintedwith the basic concepts pertaining to sheaf theory,homologicalalgebra and complex differential geometry. In the final chapte,some very recent questio and open problems are addressed, forexample results related to the finiteness of the canonical ring andthe abundance conjecture, as well as results describing thegeometric structure of Kahler varieties and their positivecones.
IntroductionChapter 1. Preliminary Material: Cohomology, Currents1.A. Dolbeault Cohomology and Sheaf Cohomology1.B. Plurisuhharmonic Functio1.C. Positive CurrentsChapter 2. Lelong numbe and Inteection Theory2.A. Multiplication of Currents and Monge-Ampere Operato2.B. Lelong NumbeChapter 3. Hermitian Vector Bundles,Connectio and CurvatureChapter 4. Bochner Technique and Vanishing Theorems4.A. Laplace-Beltrami Operato and Hodge Theory4.B. Serre Duality Theorem4.CBochner-Kodaira-Nakano Identity on Kahler Manifolds4.D. Vanishing TheoremsChapter 5. L2 Estimates and Existence Theorems5.A. Basic L2 Existence Theorems5.B. Multiplier Ideal Sheaves and Nadel Vanishing TheoremChapter 6. Numerically Effective andPseudo-effective Line Bundles6.A. Pseudo-effective Line Bundles and Metrics with MinimalSingularities6.B. Nef Line Bundles6.C. Description of the Positive Cones6.D. The Kawamat~-Viehweg Vanishing Theorem6.E. A Uniform Global Generation Property due to Y.T. SiuChapter 7. A Simple Algebraic Approach to Fujita's ConjectureChapter 8. Holomorphic Moe Inequalities8.A. General Analytic Statement on Compact Complex Manifolds8.B. Algebraic Counterparts of the Holomorphic Moe Inequalities8.C. Asymptotic Cohomology Groups8.D. Tracendental Asymptotic Cohomology FunctioChapter 9. Effective Veion of Matsusaka's Big TheoremChapter 10. Positivity Concepts for Vector BundlesChapter 11. Skoda's L2 Estimates for Surjective Bundle Morphisms11.A. Surjectivity and Division Theorems11.B. Applicatio to Local Algebra: the Brianqon-Skoda TheoremChapter 12. The Ohsawa-Takegoshi L2 Exteion Theorem12.A. The Basic a Priori Inequality12.B. Abstract L2 Existence Theorem for Solutio of O-Equatio12.C. The L2 Exteion Theorem12.D. Skoda's Division Theorem for Ideals of Holomorphic FunctioChapter 13. Approximation of Closed Positive Currentsby Analytic Cycles13.A. Approximation of Plurisubharmonic Functio Via Bergman kernels13.B. Global Approximation of Closed (1,1)-Currents on a CompactComplex Manifold13.C. Global Approximation by Diviso13.D. Singularity Exponents and log Canonical Thresholds13.E. Hodge Conjecture and approximation of (p, p)- currentsChapter 14. Subadditivity of Multiplier Idealsand Fujita's Approximate Zariski DecompositionChapter 15. Hard Lefschetz Theoremwith Multiplier Ideal Sheaves15.A. A Bundle Valued Hard Lefschetz Theorem15.B. Equisingular Approximatio of Quasi Plurisubharmonic Functio15.C. A Bochner Type Inequality15.D. Proof of Theorem 15.115.E. A CounterexampleChapter 16. Invariance of Plurigenera of Projective VarietiesChapter 17. Numerical Characterization of the K~ihler Cone17.A. Positive Classes in Intermediate (p, p)-bidegrees17.B. Numerically Positive Classes of Type (1,1)17.C. Deformatio of Compact K~hler ManifoldsChapter 18. Structure of the Pseudo-effective Coneand Mobile Inteection Theory18.A. Classes of Mobile Curves and of Mobile (n- 1, n-1)-currents18.B. Zariski Decomposition and Mobile Inteectio18.C. The Orthogonality Estimate18.D. Dual of the Pseudo-effective Cone18.E. A Volume Formula for Algebraic (1,1)-Classes on ProjectiveSurfacesChapter 19. Super-canonical Metrics and Abundance19.A. Cotruction of Super-canonical Metrics19.B. Invariance of Plurigenera and Positivity of Curvature ofSuper-canonical Metrics19.C. Tsuji's Strategy for Studying AbundanceChapter 20. Siu's Analytic Approach and Paun'sNon Vanishing TheoremReferences
In the dictionary between analytic geometry and algebraic geometry, the ideal plays a very important role, since it directly converts an analytic objectinto an algebraic one, and, simultaneously, takes care of the singularities in avery efficient way. Another analytic tool used to deal with singularities is thetheory of positive currents introduced by Lelong [Lel57]. Currents can be seen asgeneralizations of algebraic cycles, and many classical resultS of intersection theorystill apply to currents. The concept of Lelong number of a current is the analyticanalogue of the concept of multiplicity of a germ of algebraic variety. Intersectionsof cycles correspond to wedge products of currents （whenever these products aredefined）.
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