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A Survey of the Hodge Conjecture: Second Edition
James D. Lewis, University of Alberta, Edmonton, AB, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Monograph Series
1999; 368 pp; hardcover
Volume: 10
ISBN-10: 0-8218-0568-1
ISBN-13: 978-0-8218-0568-8
List Price: US$91
Member Price: US$72.80
Order Code: CRMM/10
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This book provides an introduction to a topic of central interest in transcendental algebraic geometry: the Hodge conjecture. Consisting of 15 lectures plus addenda and appendices, the volume is based on a series of lectures delivered by Professor Lewis at the Centre de Recherches Mathématiques (CRM).

The book is a self-contained presentation, completely devoted to the Hodge conjecture and related topics. It includes many examples, and most results are completely proven or sketched. The motivation behind many of the results and background material is provided. This comprehensive approach to the book gives it a "user-friendly" style. Readers need not search elsewhere for various results. The book is suitable for use as a text for a topics course in algebraic geometry; includes an appendix by B. Brent Gordon.

Titles in this series are co-published with the Centre de Recherches Mathématiques.


Graduate students and research mathematicians working in transcendental methods and Hodge theory; mathematical physicists working on Calabi-Yau manifolds, mirror symmetry or quantum cohomology.


"The first edition of this comprehensive monograph was published in 1991. Over the past 8 years, this masterly written text has become one of the most frequently used sources for geometers dealing with the subject, and it has proved to be an excellent introduction to the general framework of transcendental algebraic geometry just as well. There was and is certainly a need for such a book. This second edition of J. D. Lewis's monograph appears as an appropriately updated version of the already well-proved original text, with the advantage of being presented in a modern, more user-friendly type-setting."

-- Zentralblatt MATH

Table of Contents

  • Complex manifolds
  • Vector bundles
  • Kähler manifolds
  • Line bundles
  • The Lefschetz (1,1) theorem
  • The Lefschetz (1,1) theorem revisited
  • Formulation of the general Hodge conjecture
  • Chern class theory
  • Cohomology of complete intersections
  • The Hodge theorem
  • Analytic and topological necessities of the Kähler condition
  • Intermediate Jacobians
  • Various approaches to the Hodge conjecture for varieties with well understood geometric structure
  • The approach to the Hodge conjecture via normal functions
  • Hodge theory and Chow groups
  • Results and formulations in the singular case
  • A survey of the Hodge conjecture for abelian varieties
  • Index
  • Index of notation
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