Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals 代数几何学中的正性 II

作者： R.K. Lazarsfeld著

出 版 社： 北京燕山出版社

出版时间： 2004-10-1字数：版次： 1页数： 385印刷时间： 2004/10/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540225317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.

Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".

目录

Notation and Conventions

Part Two: Positivity for Vector Bundles

Introduction to Part Two

6 Ample and Nef Vector Bundles

6.1 Classical Theory

6.1.A Definition and First Properties

6.1.B Cohomological Properties

6.1.C Criteria for Amplitude

6.1.D Metric Approaches to Positivity of Vector Bundles

6.2 Q-Twisted and Nef Bundles

6.2.A Twists by Q-Divisors

6.2.B Nef Bundles

6.3 E~amples and Constructions

6.3.A Normal and Tangent Bundles

6.3.B Ample Cotangent Bundles and Hyperbolicity

6.3.C Picard Bundles

6.3.D The Bundle Associated to a Branched Covering..

6.3.E Direct Images of Canonical Bundles

6.3.F Some Constructions of Positive Vector Bundles ..

6.4 Ample Vector Bundles on Curves

6.4.A Review of Semistability

6.4.B Semistability and Amplitude

Notes

7 Geometric Properties of Ample Bundles

7.1 Topology

7.1.A Sommese's Theorem

7.1.B Theorem of Bloch and Gieseker

7.1.C A Barth-Type Theorem for Branched Coverings

7.2 Degeneracy Loci

7.2.A Statements and First Examples

7.2.B Proof of Connectedness of Degeneracy Loci ...

7.2.C Some Applications

7.2.D Variants and Extensions

7.3 Vanishing Theorems

7.3.A Vanishing Theorems of Griffiths and Le Potier.

7.3.B Generalizations

Notes

8 Numerical Properties of Ample Bundles

8.1 Preliminaries from Intersection Theory

8.1.A Chern Classes for Q-Twisted Bundles

8.1.B Cone Classes

8.1.C Cone Classes for Q-Twists

8.2 Positivity Theorems

8.2.A Positivity of Chern Classes

8.2.B Positivity of Cone Classes

8.3 Positive Polynomials for Ample Bundles

8.4 Some Applications

8.4.A Positivity of Intersection Products

8.4.B Non-Emptiness of Degeneracy Loci

8.4.C Singularities of Hypersurfaces Along a Curve

Notes

Part Three: Multiplier Ideals and Their Applications

Introduction to Part Three

9 Multiplier Ideal Sheaves

9.1 Preliminaries

9.1.A Q-Divisors

9.1.B Normal Crossing Divisors and Log Resolutions

9.1.C The Kawamata Viehweg Vanishing Theorem .

9.2 Definition and First Properties

9.2.A Definition of Multiplier Ideals

9.2.B First Properties

9.3 Examples and Complements

9.3.A Multiplier Ideals and Multiplicity

9.3.B Invariants Arising from Multiplier Ideals

9.3.C Monomial Ideals

9.3.D Analytic Construction of Multiplier Ideals

……

10 Some A pplications of Multiplier Ideals

11 Asymptotic Constructions

References

Glossary of Notation

Index