Positivity In Algebraic Geometry II

by Lazarsfeld, Robert

• ISBN13: 9783540225317
• ISBN10: 3540225315
• Format: Paperback
• Copyright: 2004-10-25
• Publisher: Springer Verlag

Summary

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".

Table of Contents

Notation and Conventions	1	(4)
Part Two: Positivity for Vector Bundles
Introduction to Part Two	5	(2)
Ample and Nef Vector Bundles	7	(58)
Classical Theory	7	(13)
Definition and First Properties	8	(3)
Cohomological Properties	11	(4)
Criteria for Amplitude	15	(3)
Metric Approaches to Positivity of Vector Bundles	18	(2)
Q-Twisted and Nef Bundles	20	(7)
Twists by Q-Divisors	20	(4)
Nef Bundles	24	(3)
Examples and Constructions	27	(29)
Normal and Tangent Bundles	27	(9)
Ample Cotangent Bundles and Hyperbolicity	36	(8)
Picard Bundles	44	(3)
The Bundle Associated to a Branched Covering	47	(4)
Direct Images of Canonical Bundles	51	(2)
Some Constructions of Positive Vector Bundles	53	(3)
Ample Vector Bundles on Curves	56	(9)
Review of Semistability	57	(3)
Semistability and Amplitude	60	(4)
Notes	64	(1)
Geometric Properties of Ample Bundles	65	(36)
Topology	65	(9)
Sommese's Theorem	65	(3)
Theorem of Bloch and Gieseker	68	(3)
A Barth-Type Theorem for Branched Coverings	71	(3)
Degeneracy Loci	74	(15)
Statements and First Examples	74	(4)
Proof of Connectedness of Degeneracy Loci	78	(4)
Some Applications	82	(5)
Variants and Extensions	87	(2)
Vanishing Theorems	89	(12)
Vanishing Theorems of Griffiths and Le Potier	89	(6)
Generalizations	95	(3)
Notes	98	(3)
Numerical Properties of Ample Bundles	101	(38)
Preliminaries from Intersection Theory	101	(10)
Chern Classes for Q-Twisted Bundles	102	(2)
Cone Classes	104	(6)
Cone Classes for Q-Twists	110	(1)
Positivity Theorems	111	(6)
Positivity of Chern Classes	111	(3)
Positivity of Cone Classes	114	(3)
Positive Polynomials for Ample Bundles	117	(8)
Some Applications	125	(10)
Positivity of Intersection Products	125	(2)
Non-Emptiness of Degeneracy Loci	127	(2)
Singularities of Hypersurfaces Along a Curve	129	(3)
Notes	132	(3)
Part Three: Multiplier Ideals and Their Applications
Introduction to Part Three	135	(4)
Multiplier Ideal Sheaves	139	(94)
Preliminaries	140	(11)
Q-Divisors	140	(2)
Normal Crossing Divisors and Log Resolutions	142	(5)
The Kawamata--Viehweg Vanishing Theorem	147	(4)
Definition and First Properties	151	(11)
Definition of Multiplier Ideals	152	(6)
First Properties	158	(4)
Examples and Complements	162	(23)
Multiplier Ideals and Multiplicity	162	(3)
Invariants Arising from Multiplier Ideals	165	(5)
Monomial Ideals	170	(6)
Analytic Construction of Multiplier Ideals	176	(3)
Adjoint Ideals	179	(2)
Multiplier and Jacobian Ideals	181	(1)
Multiplier Ideals on Singular Varieties	182	(3)
Vanishing Theorems for Multiplier Ideals	185	(10)
Local Vanishing for Multiplier Ideals	186	(2)
The Nadel Vanishing Theorem	188	(3)
Vanishing on Singular Varieties	191	(1)
Nadel's Theorem in the Analytic Setting	192	(1)
Non-Vanishing and Global Generation	193	(2)
Geometric Properties of Multiplier Ideals	195	(21)
Restrictions of Multiplier Ideals	195	(6)
Subadditivity	201	(3)
The Summation Theorem	204	(6)
Multiplier Ideals in Families	210	(3)
Coverings	213	(3)
Skoda's Theorem	216	(17)
Integral Closure of Ideals	216	(5)
Skoda's Theorem: Statements	221	(5)
Skoda's Theorem: Proofs	226	(2)
Variants	228	(2)
Notes	230	(3)
Some Applications of Multiplier Ideals	233	(36)
Singularities	233	(6)
Singularities of Projective Hypersurfaces	233	(2)
Singularities of Theta Divisors	235	(3)
A Criterion for Separation of Jets of Adjoint Series	238	(1)
Matsusaka's Theorem	239	(7)
Nakamaye's Theorem on Base Loci	246	(5)
Global Generation of Adjoint Linear Series	251	(11)
Fujita Conjecture and Angehrn--Siu Theorem	252	(2)
Loci of Log-Canonical Singularities	254	(4)
Proof of the Theorem of Angehrn and Siu	258	(4)
The Effective Nullstellensatz	262	(7)
Notes	267	(2)
Asymptotic Constructions	269	(54)
Construction of Asymptotic Multiplier Ideals	270	(12)
Complete Linear Series	270	(6)
Graded Systems of Ideals and Linear Series	276	(6)
Properties of Asymptotic Multiplier Ideals	282	(11)
Local Statements	282	(3)
Global Results	285	(7)
Multiplicativity of Plurigenera	292	(1)
Growth of Graded Families and Symbolic Powers	293	(6)
Fujita's Approximation Theorem	299	(13)
Statement and First Consequences	299	(6)
Proof of Fujita's Theorem	305	(2)
The Dual of the Pseudoeffective Cone	307	(5)
Siu's Theorem on Plurigenera	312	(11)
Notes	320	(3)
References	323	(34)
Glossary of Notation	357	(6)
Index	363