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| Notation and Conventions Part Two: Positivity for Vector Bundles Introduction to Part Two 6 Ample and Nef Vector Bundles 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 E~amples 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 Part Three: Multiplier Ideals and Their Applications Introduction to Part Three 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 …… 10 Some A pplications of Multiplier Ideals 11 Asymptotic Constructions References Glossary of Notation Index |
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