Hodge Theory

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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 structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students. Read more... Abstract: 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 structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students Content: 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 Étalé 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 Poincaré 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.