Positivity In Algebraic Geometry I

by Lazarsfeld, Robert

• ISBN13: 9783540225287
• ISBN10: 3540225285
• eBook ISBN(s): 9783642188084
• Format: Paperback
• Copyright: 2004-10-25
• Publisher: Springer Verlag
• More Book Details
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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.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".

Table of Contents

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