# Introduction to Hodge Theory

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*José Bertin; Jean-Pierre Demailly; Luc Illusie; Chris Peters*

A co-publication of the AMS and Société Mathématique de France

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 & Endorsements

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

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