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Introduction to Hodge Theory
José Bertin, University of Grenoble I, St. Martin D'Heres, France, Jean-Pierre Demailly, University of Grenoble I, St. Martin d'Heres, France, Luc Illusie, University of Paris-Sud, Orsay, France, and Chris Peters, University of Grenoble I, St. Martin d'Heres, France
A co-publication of the AMS and Société Mathématique de France.
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SMF/AMS Texts and Monographs
2002; 232 pp; softcover
Volume: 8
ISBN-10: 0-8218-2040-0
ISBN-13: 978-0-8218-2040-7
List Price: US$75
Member Price: US$60
Order Code: SMFAMS/8
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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.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students, research mathematicians, and physicists interested in Hodge theory.

Reviews

"This profound introduction to classical and modern Hodge theory, which discusses the subject in great depth and leads the reader to the forefront of contemporary research in many areas related to Hodge theory ... a masterly guide through Hodge theory and its various applications ... its significant role as an indispensible source for active researchers and teachers in the field ... its translation into English makes it now accessible to the entire mathematical and physical community worldwide. Without any doubt, this is exactly what both those communities and this excellent book on Hodge theory needed and deserved."

-- Zentralblatt MATH

From reviews of the French Edition:

"The present book ... may be regarded as a masterly introduction to Hodge theory in its classical and very recent, analytic and algebraic aspects ... it is by far much more than only an introduction to the subject. The material leads the reader to the forefront of research in many areas related to Hodge theory, and that in a detailed highly self-contained manner ... this text is also a valuable source for active researchers and teachers in the field ..."

-- Zentralblatt MATH

"The book under review is a collection of three articles about Hodge theory and related developments, which are all aimed at non-experts and fulfill, in an extremely satisfactory manner, two functions. First, the basic methods used in the theories are discussed and developed in great detail; second, some newer developments are described, giving the reader a good overview of the more important applications. Furthermore, the style makes these articles a joy to work through, even for the mathematician not encountering these subjects for the first time."

-- Mathematical Reviews

Table of Contents

  • J.-P. Demailly -- \(L^2\) Hodge theory and vanishing theorems
  • L. Illusie -- Frobenius and Hodge degeneration
  • J. Bertin and C. Peters -- Variations of Hodge structure, Calabi-Yau manifolds and mirror symmetry
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