Positivity In Algebraic Geometry I

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".

Notation and Conventions

1

(4)

Part One: Ample Line Bundles and Linear Series

Introduction to Part One

5

(2)

Ample and Nef Line Bundles

7

(114)

Preliminaries: Divisors, Line Bundles, and Linear Series

7

(17)

Divisors and Line Bundles

8

(4)

Linear Series

12

(3)

Intersection Numbers and Numerical Equivalence

15

(5)

Riemann--Roch

20

(4)

The Classical Theory

24

(20)

Cohomological Properties

25

(8)

Numerical Properties

33

(6)

Metric Characterizations of Amplitude

39

(5)

Q-Divisors and R-Divisors

44

(6)

Definitions for Q-Divisors

44

(4)

R-Divisors and Their Amplitude

48

(2)

Nef Line Bundles and Divisors

50

(20)

Definitions and Formal Properties

51

(2)

Kleiman's Theorem

53

(6)

Cones

59

(6)

Fujita's Vanishing Theorem

65

(5)

Examples and Complements

70

(18)

Ruled Surfaces

70

(3)

Products of Curves

73

(6)

Abelian Varieties

79

(1)

Other Varieties

80

(2)

Local Structure of the Nef Cone

82

(4)

The Cone Theorem

86

(2)

Inequalities

88

(6)

Global Results

88

(3)

Mixed Multiplicities

91

(3)

Amplitude for a Mapping

94

(4)

Castelnuovo--Mumford Regularity

98

(23)

Definitions, Formal Properties, and Variants

99

(8)

Regularity and Complexity

107

(3)

Regularity Bounds

110

(5)

Syzygies of Algebraic Varieties

115

(4)

Notes

119

(2)

Linear Series

121

(64)

Asymptotic Theory

121

(18)

Basic Definitions

122

(6)

Semiample Line Bundles

128

(5)

Iitaka Fibration

133

(6)

Big Line Bundles and Divisors

139

(18)

Basic Properties of Big Divisors

139

(6)

Pseudoeffective and Big Cones

145

(3)

Volume of a Big Divisor

148

(9)

Examples and Complements

157

(15)

Zariski's Construction

158

(1)

Cutkosky's Construction

159

(5)

Base Loci of Nef and Big Linear Series

164

(2)

The Theorem of Campana and Peternell

166

(1)

Zariski Decompositions

167

(5)

Graded Linear Series and Families of Ideals

172

(13)

Graded Linear Series

172

(4)

Graded Families of Ideals

176

(7)

Notes

183

(2)

Geometric Manifestations of Positivity

185

(54)

The Lefschetz Theorems

185

(16)

Topology of Affine Varieties

186

(6)

The Theorem on Hyperplane Sections

192

(7)

Hard Lefschetz Theorem

199

(2)

Projective Subvarieties of Small Codimension

201

(6)

Barth's Theorem

201

(3)

Hartshorne's Conjectures

204

(3)

Connectedness Theorems

207

(6)

Bertini Theorems

207

(3)

Theorem of Fulton and Hansen

210

(2)

Grothendieck's Connectedness Theorem

212

(1)

Applications of the Fulton--Hansen Theorem

213

(18)

Singularities of Mappings

214

(5)

Zak's Theorems

219

(8)

Zariski's Problem

227

(4)

Variants and Extensions

231

(8)

Homogeneous Varieties

231

(2)

Higher Connectivity

233

(4)

Notes

237

(2)

Vanishing Theorems

239

(30)

Preliminaries

240

(8)

Normal Crossings and Resolutions of Singularities

240

(2)

Covering Lemmas

242

(6)

Kodaira and Nakano Vanishing Theorems

248

(4)

Vanishing for Big and Nef Line Bundles

252

(9)

Statement and Proof of the Theorem

252

(5)

Some Applications

257

(4)

Generic Vanishing Theorem

261

(8)

Notes

267

(2)

Local Positivity

269

(56)

Seshadri Constants

269

(9)

Lower Bounds

278

(12)

Background and Statements

278

(4)

Multiplicities of Divisors in Families

282

(4)

Proof of Theorem 5.2.5

286

(4)

Abelian Varieties

290

(13)

Period Lengths and Seshadri Constants

290

(7)

Proof of Theorem 5.3.6

297

(4)

Complements

301

(2)

Local Positivity Along an Ideal Sheaf

303

(12)

Definition and Formal Properties of the s-Invariant

303

(5)

Complexity Bounds

308

(4)

Notes

312

(3)

Appendices

A Projective Bundles

315

(2)

B Cohomology and Complexes

317

(8)

B.1 Cohomology

317

(3)

B.2 Complexes

320

(5)

References

325

(34)

Glossary of Notation

359

(6)

Index

365

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