CRM Monograph Series 1999; 368 pp; hardcover Volume: 10 ISBN10: 0821805681 ISBN13: 9780821805688 List Price: US$96 Member Price: US$76.80 Order Code: CRMM/10
 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 selfcontained 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 "userfriendly" 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 copublished with the Centre de Recherches Mathématiques. Readership Graduate students and research mathematicians working in transcendental methods and Hodge theory; mathematical physicists working on CalabiYau manifolds, mirror symmetry or quantum cohomology. Reviews "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 wellproved original text, with the advantage of being presented in a modern, more userfriendly typesetting."  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
