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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
Chris Peters University of Grenoble I, St. Martin d’Heres, France
A co-publication of the AMS and the Société Mathématique de France
Available Formats:
Softcover ISBN: 978-0-8218-2040-7
Product Code: SMFAMS/8
232 pp
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Click above image for expanded view 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 Chris Peters University of Grenoble I, St. Martin d’Heres, France A co-publication of the AMS and the Société Mathématique de France Available Formats:  Softcover ISBN: 978-0-8218-2040-7 Product Code: SMFAMS/8 232 pp  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
• Book Details

SMF/AMS Texts and Monographs
Volume: 82002
MSC: Primary 14; 13; 58; 32; Secondary 81;

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.

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
• Request Review Copy
Volume: 82002
MSC: Primary 14; 13; 58; 32; Secondary 81;

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.

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

• 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
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