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Viser: Positivity in Algebraic Geometry - Positivity for Vector Bundles, and Multiplier Ideals
Positivity in Algebraic Geometry
Positivity for Vector Bundles, and Multiplier Ideals
R. K. Lazarsfeld og University of Michigan Staff
(2004)
Sprog: Engelsk
Detaljer om varen
- 1. Udgave
- Hardback: 385 sider
- Udgiver: Springer Berlin / Heidelberg (August 2004)
- Forfattere: R. K. Lazarsfeld og University of Michigan Staff
- ISBN: 9783540225348
Both volumes also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".
A good deal of the material has not previously appeared in book form.
Volume II is more at the research level and somewhat more specialized than Volume I.
Volume II contains a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications.
Contains many concrete examples, applications, and pointers to further developments
6.1 Classical Theory.-
6.1.A Definition and First Properties.-
6.1.B Cohomological Properties.-
6.1.C Criteria for Amplitude.-
6.1.D Metric Approaches to Positivity of Vector Bundles.-
6.2 Q-Twisted and Nef Bundles.-
6.2.A Twists by Q-Divisors.-
6.2.B Nef Bundles.-
6.3 Examples and Constructions.-
6.3.A Normal and Tangent Bundles.-
6.3.B Ample Cotangent Bundles and Hyperbolicity.-
6.3.C Picard Bundles.-
6.3.D The Bundle Associated to a Branched Covering.-
6.3.E Direct Images of Canonical Bundles.-
6.3.F Some Constructions of Positive Vector Bundles.-
6.4 Ample Vector Bundles on Curves.-
6.4.A Review of Semistability.-
6.4.B Semistability and Amplitude.- Notes.- 7 Geometric Properties of Ample Bundles.-
7.1 Topology.-
7.1.A Sommese''s Theorem.-
7.1.B Theorem of Bloch and Gieseker.-
7.1.C A Barth-Type Theorem for Branched Coverings.-
7.2 Degeneracy Loci.-
7.2.A Statements and First Examples.-
7.2.B Proof of Connectedness of Degeneracy Loci.-
7.2.C Some Applications.-
7.2.D Variants and Extensions.-
7.3 Vanishing Theorems.-
7.3.A Vanishing Theorems of Griffiths and Le Potier.-
7.3.B Generalizations.- Notes.- 8 Numerical Properties of Ample Bundles.-
8.1 Preliminaries from Intersection Theory.-
8.1.A Chern Classes for Q-Twisted Bundles.-
8.1.B Cone Classes.-
8.1.C Cone Classes for Q-Twists.-
8.2 Positivity Theorems.-
8.2.A Positivity of Chern Classes.-
8.2.B Positivity of Cone Classes.-
8.3 Positive Polynomials for Ample Bundles.-
8.4 Some Applications.-
8.4.A Positivity of Intersection Products.-
8.4.B Non-Emptiness of Degeneracy Loci.-
8.4.C Singularities of Hypersurfaces Along a Curve.- Notes.- Three: Multiplier Ideals and Their Applications.- 9 Multiplier Ideal Sheaves.-
9.1 Preliminaries.-
9.1.A Q-Divisors.-
9.1.B Normal Crossing Divisors and Log Resolutions.-
9.1.C The Kawamata--Viehweg Vanishing Theorem.-
9.2 Definition and First Properties.-
9.2.A Definition of Multiplier Ideals.-
9.2.B First Properties.-
9.3 Examples and Complements.-
9.3.A Multiplier Ideals and Multiplicity.-
9.3.B Invariants Arising from Multiplier Ideals.-
9.3.C Monomial Ideals.-
9.3.D Analytic Construction of Multiplier Ideals.-
9.3.E Adjoint Ideals.-
9.3.F Multiplier and Jacobian Ideals.-
9.3.G Multiplier Ideals on Singular Varieties.-
9.4 Vanishing Theorems for Multiplier Ideals.-
9.4.A Local Vanishing for Multiplier Ideals.-
9.4.B The Nadel Vanishing Theorem.-
9.4.C Vanishing on Singular Varieties.-
9.4.D Nadel''s Theorem in the Analytic Setting.-
9.4.E Non-Vanishing and Global Generation.-
9.5 Geometric Properties of Multiplier Ideals.-
9.5.A Restrictions of Multiplier Ideals.-
9.5.B Subadditivity.-
9.5.C The Summation Theorem.-
9.5.D Multiplier Ideals in Families.-
9.5.E Coverings.-
9.6 Skoda''s Theorem.-
9.6.A Integral Closure of Ideals.-
9.6.B Skoda''s Theorem: Statements.-
9.6.C Skoda''s Theorem: Proofs.-
9.6.D Variants.- Notes.- 10 Some Applications of Multiplier Ideals.-
10.1 Singularities.-
10.1.A Singularities of Projective Hypersurfaces.-
10.1.B Singularities of Theta Divisors.-
10.1.C A Criterion for Separation of Jets of Adjoint Series.-
10.2 Matsusaka''s Theorem.-
10.3 Nakamaye''s Theorem on Base Loci.-
10.4 Global Generation of Adjoint Linear Series.-
10.4.A Fujita Conjecture and Angehrn--Siu Theorem.-
10.4.B Loci of Log-Canonical Singularities.-
10.4.C Proof of the Theorem of Angehrn and Siu.-
10.5 The Effective Nullstellensatz.- Notes.- 11 Asymptotic Constructions.-
11.1 Construction of Asymptotic Multiplier Ideals.-
11.1.A Complete Linear Series.-
11.1.B Graded Systems of Ideals and Linear Series.-
11.2 Properties of Asymptotic Multiplier Ideals.-
11.2.A Local Statements.-
11.2.B Global Results.-
11.2.C Multiplicativity of Plurigenera.-
11.3 Growth of Graded Families and Symbolic Powers.-
11.4 Fujita''s Approximation Theorem.-
11.4.A Statement and First Consequences.-
11.4.B Proof of Fujita''s Theorem.-
11.4.C The Dual of the Pseudoeffective Cone.-
11.5.- Notes.- References.- Glossary of Notation.