Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series

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Springer Science & Business Media, Aug 24, 2004 - Mathematics - 387 pages
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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.

 

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Contents

Ample and Nef Line Bundles
7
11A Divisors and Line Bundles
8
11B Linear Series
12
11C Intersection Numbers and Numerical Equivalence
15
11D RiemannRoch
20
12 The Classical Theory
24
12A Cohomological Properties
25
12B Numerical Properties
33
24B Graded Families of Ideals
176
Notes
183
Geometric Manifestations of Positivity
185
31A Topology of Affine Varieties
186
31B The Theorem on Hyperplane Sections
192
31C Hard Lefschetz Theorem
199
32 Projective Subvarieties of Small Codimension
201
32B Hartshornes Conjectures
204

12C Metric Characterizations of Amplitude
39
13 QDivisors and RDivisors
44
13B RDivisors and Their Amplitude
48
14 Nef Line Bundles and Divisors
50
14A Definitions and Formal Properties
51
14B Kleimans Theorem
53
14C Cones
59
14D Fujitas Vanishing Theorem
65
15 Examples and Complements
70
15B Products of Curves
73
15C Abelian Varieties
79
15D Other Varieties
80
15E Local Structure of the Nef Cone
82
15F The Cone Theorem
86
16 Inequalities
88
16B Mixed Multiplicities
91
17 Amplitude for a Mapping
94
18 CastelnuovoMumford Regularity
98
18A Definitions Formal Properties and Variants
99
18B Regularity and Complexity
107
18C Regularity Bounds
110
18D Syzygies of Algebraic Varieties
115
Notes
119
Linear Series
121
21A Basic Definitions
122
21B Semiample Line Bundles
128
21C litaka Fibration
133
22 Big Line Bundles and Divisors
139
22B Pseudoeffective and Big Cones
145
22C Volume of a Big Divisor
148
23 Examples and Complements
157
23A Zariskis Construction
158
23B Cutkoskys Construction
159
23C Base Loci of Nef and Big Linear Series
164
23D The Theorem of Campana and Peternell
166
23E Zariski Decompositions
167
24 Graded Linear Series and Families of Ideals
172
33 Connectedness Theorems
207
33B Theorem of Fulton and Hansen
210
33C Grothendiecks Connectedness Theorem
212
34 Applications of the FultonHansen Theorem
213
34A Singularities of Mappings
214
34B Zaks Theorems
219
34C Zariskis Problem
227
35 Variants and Extensions
231
35B Higher Connectivity for Mappings to Projective Space
233
Notes
237
Vanishing Theorems
239
41 Preliminaries
240
41B Covering Lemmas
242
42 Kodaira and Nakano Vanishing Theorems
248
43 Vanishing for Big and Nef Line Bundles
252
43B Some Applications
257
44 Generic Vanishing Theorem
261
Notes
267
Local Positivity
269
52 Lower Bounds
278
52B Multiplicities of Divisors in Families
282
52C Proof of Theorem 525
286
53 Abelian Varieties
290
53B Proof of Theorem 536
297
53C Complements
301
54 Local Positivity Along an Ideal Sheaf
303
54B Complexity Bounds
308
Notes
312
Appendices
313
Projective Bundles
315
Cohomology and Complexes
317
B2 Complexes
320
References
325
Glossary of Notation
359
Index
365
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