Positivity in algebraic geometry I : classical setting :,line bundles and linear series代数几何学中的正性 I

作者： R.K. Lazarsfeld 著

出 版 社： 北京燕山出版社

出版时间： 2007-5-1字数：版次： 1页数： 387印刷时间： 2007/05/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540225287包装： 平装内容简介

This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments.

Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II. Both volumes are also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".

目录

Contents

Notation and Conventions

Part One: Ample Line Bundles and Linear Series

Introduction to Part One

1 Ample and Nef Line Bundles

1.1 Preliminaries: Divisors, Line Bundles, and Linear Series

1.1.A Divisors and Line Bundles

1.1.B Linear Series

1.1.C Intersection Numbers and Numerical Equivalence ..

1.1.D Riemann-Roch

1.2 The Classical Theory

1.2.A Cohomological Properties

1.2.B Numerical Properties

1.2.C Metric Characterizations of Amplitude

1.3 Q-Divisors and R-Divisors

1.3.A Definitions for Q-Divisors

1.3.B R-Divisors and Their Amplitude

1.4 Nef Line Bundles and Divisors

1.4.A Definitions and Formal Properties

1.4.B Kleiman's Theorem

1.4.C Cones

1.4.D Fujita's Vanishing Theorem

1.5 Examples and Complements

1.5.A Ruled Surfaces

1.5.B Products of Curves

1.5.C Abelian Varieties

1.5.D Other Varieties

1.5.E Local Structure of the Nef Cone

1.5.F The Cone Theorem

1.6 Inequalities

1.6.A Global Results

1.6.B Mixed Multiplicities

1.7 Amplitude for a Mapping

1.8 Castelnuovo-Mumford Regularity

1.8.A Definitions, Formal Properties, and Variants

1.8.B Regularity and Complexity

1.8.C Regularity Bounds

1.8.D Syzygies of Algebraic Varieties

Notes

2 Linear Series

2.1 Asymptotic Theory

2.1.A Basic Definitions

2.1.B Semiample Line Bundles

2.1.C Iitaka Fibration

2.2 Big Line Bundles and Divisors

2.2.A Basic Properties of Big Divisors

2.2.B Pseudoeffective and Big Cones

2.2.C Volume of a Big Divisor

2.3 Examples and Complements

2.3.A Zariski's Construction

2.3.B Cutkosky's Construction

2.3.C Base Loci of Nef and Big Linear Series

2.3.D The Theorem of Campana and Peternell

2.3.E Zariski Decompositions

2.4 Graded Linear Series and Families of Ideals

2.4.A Graded Linear Series

Notes

3 Geometric Manifestations of Positivity

3.1 The Lefschetz Theorems

3.1.A Topology of Afline Varieties

3.1.B The Theorem on Hyperplane Sections

3.1.C Hard Lefschetz Theorem

3.2 Projective Subvarieties of Small Codimension

3.2.A Barth's Theorem

3.2.B Hartshorne's Conjectures

3.3 Connectedness Theorems

3.3.A Bertini Theorems

3.3.B Theorem of Fulton and Hansen

3.3.C Grothendieck's Connectedness Theorem

3.4 Applications of the Fulton-Hansen Theorem

3.4.A Singularities of Mappings

……

4 Vanishing Theorems

5 Local Positivity

Appendices

A PROJECTIVE bUNDLES

B Cohomology and Complexes

References

Glossary of Notation

Index

Contets of VolumeⅡ

Notation and Conventios

Part Two:Positivity for Vector Bundles

Part Three:Multiplier Ideals and Their Applictions

References

Glossary of Notation

Index