Positivity in algebraic geometry 2

Front Cover
Springer Science & Business Media, Aug 24, 2004 - Mathematics - 385 pages
0 Reviews

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".

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Ample and Nef Vector Bundles
7
61A Definition and First Properties
8
61B Cohomological Properties
11
61C Criteria for Amplitude
15
61D Metric Approaches to Positivity of Vector Bundles
18
62 QTwisted and Nef Bundles
20
62B Nef Bundles
24
63 Examples and Constructions
27
93C Monomial Ideals
170
93D Analytic Construction of Multiplier Ideals
176
93E Adjoint Ideals
179
93F Multiplier and Jacobian Ideals
181
93G Multiplier Ideals on Singular Varieties
182
94 Vanishing Theorems for Multiplier Ideals
185
94A Local Vanishing for Multiplier Ideals
186
94B The Nadel Vanishing Theorem
188

63B Ample Cotangent Bundles and Hyperbolicity
36
63C Picard Bundles
44
63D The Bundle Associated to a Branched Covering
47
63E Direct Images of Canonical Bundles
51
63F Some Constructions of Positive Vector Bundles
53
64 Ample Vector Bundles on Curves
56
64A Review of Semistability
57
64B Semistability and Amplitude
60
Notes
64
Geometric Properties of Ample Bundles
65
71B Theorem of Bloch and Gieseker
68
71C A BarthType Theorem for Branched Coverings
71
72 Degeneracy Loci
74
72B Proof of Connectedness of Degeneracy Loci
78
72C Some Applications
82
72D Variants and Extensions
87
73 Vanishing Theorems
89
73B Generalizations
95
Notes
98
Numerical Properties of Ample Bundles
101
81A Chern Classes for QTwisted Bundles
102
81B Cone Classes
104
81C Cone Classes for QTwists
110
82 Positivity Theorems
111
82B Positivity of Cone Classes
114
83 Positive Polynomials for Ample Bundles
117
84 Some Applications
125
84B NonEmptiness of Degeneracy Loci
127
84C Singularities of Hypersurfaces Along a Curve
129
Notes
132
Introduction to Part Three
135
Multiplier Ideal Sheaves
139
91 Preliminaries
140
91B Normal Crossing Divisors and Log Resolutions
142
91C The KawamataViehweg Vanishing Theorem
147
92 Definition and First Properties
151
92A Definition of Multiplier Ideals
152
92B First Properties
158
93 Examples and Complements
162
93B Invariants Arising from Multiplier Ideals
165
94C Vanishing on Singular Varieties
191
94D Nadels Theorem in the Analytic Setting
192
94E NonVanishing and Global Generation
193
95 Geometric Properties of Multiplier Ideals
195
95B Subadditivity
201
95C The Summation Theorem
204
95D Multiplier Ideals in Families
210
95E Coverings
213
96 Skodas Theorem
216
Statements
221
Proofs
226
96D Variants
228
Notes
230
Some Applications of Multiplier Ideals
233
101B Singularities of Theta Divisors
235
101 C A Criterion for Separation of Jets of Adjoint Series
238
102 Matsusakas Theorem
239
103 Nakamayes Theorem on Base Loci
246
104 Global Generation of Adjoint Linear Series
251
104A Fujitas Conjecture and AngehrnSiu Theorem
252
104B Loci of LogCanonical Singularities
254
104C Proof of the Theorem of Angehrn and Siu
258
105 The Effective Nullstellensatz
262
Notes
267
Asymptotic Constructions
269
111 Construction of Asymptotic Multiplier Ideals
270
111B Graded Systems of Ideals and Linear Series
276
112 Properties of Asymptotic Multiplier Ideals
282
112B Global Results
285
112C Multiplicativity of Plurigenera
292
113 Growth of Graded Families and Symbolic Powers
293
114 Fujitas Approximation Theorem
299
114B Proof of Fujitas Theorem
305
114C The Dual of the Pseudoeffective Cone
307
115 Sius Theorem on Plurigenera
312
Notes
320
References
323
Glossary of Notation
357
Index
363
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 336 - Griffiths, Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles, J.
Page 323 - Y.-T. Siu: Effective Freeness and Point Separation for Adjoint Bundles. Invent. Math. 122 (1995) 291-308 [Artin62] M.
Page 337 - On the vanishing of local homotopy groups for isolated singularities of complex spaces, J. Reine Angew. Math. 323 (1981), 172-176.

References to this book

All Book Search results »