代數(shù)幾何中的解析方法

前言

　　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.

內(nèi)容概要

This volume is an expaion of lectures
given by the author at the Park City Mathematics Ititute in 2008 as
well as in other places. The main purpose of the book is to
describe analytic techniques which are useful to study questio such
as linear series, multiplier ideals and vanishing theorems for
algebraic vector bundles. The exposition tries to be as condeed 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 chapte,
some very recent questio 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.

書籍目錄

Introduction
Chapter 1. Preliminary Material: Cohomology, Currents
1.A. Dolbeault Cohomology and Sheaf Cohomology
1.B. Plurisuhharmonic Functio
1.C. Positive Currents
Chapter 2. Lelong numbe and Inteection Theory
2.A. Multiplication of Currents and Monge-Ampere Operato
2.B. Lelong Numbe
Chapter 3. Hermitian Vector Bundles,Connectio and Curvature
Chapter 4. Bochner Technique and Vanishing Theorems
4.A. Laplace-Beltrami Operato 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 Moe Inequalities
8.A. General Analytic Statement on Compact Complex Manifolds
8.B. Algebraic Counterparts of the Holomorphic Moe Inequalities
8.C. Asymptotic Cohomology Groups
8.D. Tracendental Asymptotic Cohomology Functio
Chapter 9. Effective Veion 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. Applicatio to Local Algebra: the Brianqon-Skoda Theorem
Chapter 12. The Ohsawa-Takegoshi L2 Exteion Theorem
12.A. The Basic a Priori Inequality
12.B. Abstract L2 Existence Theorem for Solutio of O-Equatio
12.C. The L2 Exteion Theorem
12.D. Skoda's Division Theorem for Ideals of Holomorphic Functio
Chapter 13. Approximation of Closed Positive Currents
by Analytic Cycles
13.A. Approximation of Plurisubharmonic Functio Via Bergman kernels
13.B. Global Approximation of Closed (1,1)-Currents on a Compact
Complex Manifold
13.C. Global Approximation by Diviso
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 Approximatio of Quasi Plurisubharmonic Functio
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. Deformatio of Compact K~hler Manifolds
Chapter 18. Structure of the Pseudo-effective Cone
and Mobile Inteection Theory
18.A. Classes of Mobile Curves and of Mobile (n- 1, n-1)-currents
18.B. Zariski Decomposition and Mobile Inteectio
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. Cotruction 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

章節(jié)摘錄

　　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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