Introduction to Hodge Theory

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American Mathematical Soc., 2002 - Mathematics - 232 pages
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Hodge theory originated as an application of harmonic theory to the study of the geometry of compact complex manifolds. The ideas have proved to be quite powerful, leading to fundamentally important results throughout algebraic geometry. This book consists of expositions of various aspects of modern Hodge theory. Its purpose is to provide the nonexpert reader with a precise idea of the current status of the subject. The three chapters develop distinct but closely related subjects: $L^2$ Hodge theory and vanishing theorems; Frobenius and Hodge degeneration; variations of Hodge structures and mirror symmetry.The techniques employed cover a wide range of methods borrowed from the heart of mathematics: elliptic PDE theory, complex differential geometry, algebraic geometry in characteristic $p$, cohomological and sheaf-theoretic methods, deformation theory of complex varieties, Calabi-Yau manifolds, singularity theory, etc. A special effort has been made to approach the various themes from their most natural starting points. Each of the three chapters is supplemented with a detailed introduction and numerous references. The reader will find precise statements of quite a number of open problems that have been the subject of active research in recent years. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.
 

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Contents

Introduction
5
L2 Hodge Theory and Vanishing Theorems
7
L2 Hodge theory
9
Differential operators on vector bundles
12
Fundamental results on elliptic operators I
14
Hodge theory of compact Ricmmanian manifolds
19
Hermitian and Kahler manifolds
24
Fundamental identities of Kiihlerian geometry
27
Smoothness and liftings
107
Frobenius and Cartier isomorphism
113
Derived categories and spectral sequences
119
Decomposition degeneration and vanishing theorems in characteristic p 0
124
From characteristic p 0 to characteristic zero
130
Recent developments and open problems
137
parallelizability and ordinarity
144
Bibliography
147

The groups HpqX E and Serre duality
35
Comohology of compact Kiihlcr manifolds
36
The HodgeFrolichcr spectral sequence
43
Deformations and the semicontinuity theorem 17
46
L2 estimates and vanishing theorems
53
Hodge theory of complete Kiihler manifolds
60
Bochner techniques and vanishing theorems
70
L2 estimates and existence theorems
73
Vanishing theorems of Nadel and KawamataViehweg
75
On the conjecture of Fujita
82
An effective version of Matsusakas big theorem
89
Bibliography
95
Frobenius and Hodge Degeneration Luc ILLUSIE
99
Introduction
101
differentials the de Rham complex
103
Variations of Hodge Structure CalabiYau Manifolds and Mirror Symmetry JOSE BERTIN AND CHRIS PETERS
151
Introduction
153
Variations of Hodge structures
161
GaussManin connection
163
Variation of Hodge structures
172
Degenerations
179
Higgs bundles
187
Hodge modules
188
Mirror symmetry and CalabiYau manifolds
193
Cohomology of hypersurfaces
199
PicardFuchs equations
205
CalabiYau threefolds and mirror symmetry
210
Relation with mixed Hodge theory
222
Bibliography
229
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