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Hodge Theory (MN-49)
This book provides a comprehensive and up-to-date introduction to Hodge theory-one of the central and most vibrant areas of contemporary mathematics-from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge stru...
| Otros Autores: | , , , |
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| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton :
Princeton University Press,
2014.
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| Colección: | Book collections on Project MUSE.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Contributors; Contributors; Contents; Preface; 1 Kähler Manifolds; 1.1 Complex Manifolds; 1.1.1 Definition and Examples; 1.1.2 Holomorphic Vector Bundles; 1.2 Differential Forms on Complex Manifolds; 1.2.1 Almost Complex Manifolds; 1.2.2 Tangent and Cotangent Space; 1.2.3 De Rham and Dolbeault Cohomologies; 1.3 Symplectic, Hermitian, and Kähler Structures; 1.3.1 Kähler Manifolds; 1.3.2 The Chern Class of a Holomorphic Line Bundle; 1.4 Harmonic Forms-Hodge Theorem; 1.4.1 Compact Real Manifolds; 1.4.2 The [del symbol] -Laplacian; 1.5 Cohomology of Compact Kähler Manifolds.
- 1.5.1 The Kähler Identities1.5.2 The Hodge Decomposition Theorem; 1.5.3 Lefschetz Theorems and Hodge-Riemann Bilinear Relations; A Linear Algebra; A.1 Real and Complex Vector Spaces; A.2 The Weight Filtration of a Nilpotent Transformation; A.3 Representations of sl(2,C) and Lefschetz Theorems; A.4 Hodge Structures; B The Kähler Identities; B.1 Symplectic Linear Algebra; B.2 Compatible Inner Products; B.3 Symplectic Manifolds; B.4 The Kähler Identities; Bibliography; 2 The Algebraic de Rham Theorem; Introduction; Part I. Sheaf Cohomology, Hypercohomology, and the Projective Case; 2.1 Sheaves.
- 2.1.1 The Étale Space of a Presheaf2.1.2 Exact Sequences of Sheaves; 2.1.3 Resolutions; 2.2 Sheaf Cohomology; 2.2.1 Godement's Canonical Resolution; 2.2.2 Cohomology with Coefficients in a Sheaf; 2.2.3 Flasque Sheaves; 2.2.4 Cohomology Sheaves and Exact Functors; 2.2.5 Fine Sheaves; 2.2.6 Cohomology with Coefficients in a Fine Sheaf; 2.3 Coherent Sheaves and Serre's GAGA Principle; 2.4 The Hypercohomology of a Complex of Sheaves; 2.4.1 The Spectral Sequences of Hypercohomology; 2.4.2 Acyclic Resolutions; 2.5 The Analytic de Rham Theorem; 2.5.1 The Holomorphic Poincare Lemma.
- 2.5.2 The Analytic de Rham Theorem2.6 The Algebraic de Rham Theorem for a Projective Variety; Part II. Čech Cohomology and the Algebraic de Rham Theorem in General; 2.7 Čech Cohomology of a Sheaf; 2.7.1 Čech Cohomology of an Open Cover; 2.7.2 Relation Between Čech Cohomology and Sheaf Cohomology; 2.8 Čech Cohomology of a Complex of Sheaves; 2.8.1 The Relation Between Čech Cohomology and Hypercohomology; 2.9 Reduction to the Affine Case; 2.9.1 Proof that the General Case Implies the Affine Case; 2.9.2 Proof that the Affine Case Implies the General Case.
- 2.10 The Algebraic de Rham Theorem for an Affine Variety2.10.1 The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms; 2.10.2 The Hypercohomology of Rational and Meromorphic Forms; 2.10.3 Comparison of Meromorphic and Smooth Forms; Bibliography; 3 Mixed Hodge Structures; 3.1 Hodge Structure on a Smooth Compact Complex Variety; 3.1.1 Hodge Structure (HS); 3.1.2 Spectral Sequence of a Filtered Complex; 3.1.3 Hodge Structure on the Cohomology of Nonsingular Compact Complex Algebraic Varieties; 3.1.4 Lefschetz Decomposition and Polarized Hodge Structure; 3.1.5 Examples.