BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 4, Pages 507–520 S 0273-0979(05)01056-6 Article electronically published on April 7, 2005 Hodge theory and complex algebraic geometry I, II, by Claire Voisin, Cambridge University Press, New York, 2002, ix+322 pp., £65, ISBN 0-521-80260-1 (Vol. I); 2003, ix+351 pp., £65, ISBN 0-521-80283-0 (Vol. II)
1. Introduction The task of reviewing Claire Voisin’s two-volume work Hodge Theory and Com- plex Algebraic Geometry [V] is a daunting one, given the scope of the subject matter treated, namely, a rather complete tour of the subject from the beginning to the present, and given the break-neck pace of Voisin’s clear, complete, but “take no prisoners” exposition. As is the case with most substantial mathematical treatises, digesting the content of these volumes can only occur by reconstructing and reorga- nizing their material for oneself, led forward, of course, by the clear beacon carried by the author, one of the foremost leaders in the field. Rather than talk descrip- tively about the mathematics, I have chosen to exemplify below a strand of my own adventure in understanding and reconstruction as I worked my way through the two volumes with a group of graduate students. I hope this will serve as an invitation to other readers to do likewise. Their efforts will be amply rewarded. I beg my readers’ indulgence for the longer-than-usual review that resulted. As Voisin does, I have tried to write in such a way that readers of varying levels of background or interest can enter or leave the exposition as suits their needs and still come away with a mathematical experience of some integrity and completeness. And, as Voisin does, we start at the beginning.
2. deRham cohomology The most natural and fundamental objects in geometry are the differentiable manifolds, that is, topological spaces which are everywhere ‘in microcosm’ like Eu- clidean space in a consistent way as one moves from point to point. The differen- tiable structure on M is uniquely characterized by specifying which locally defined functions on M are the differentiable ones. The strongest form that a differential structure can take is to specify in a consistent way the functions which have a con- vergent Taylor series, the real analytic functions. Pioneering work of Whitney and Nash showed that every differentiable structure could be, in an appropriate sense, approximated by a real analytic one and that, again in an appropriate sense, real analytic structures had only trivial (infinitesimal) deformations. Early on, the study of the differential forms on a manifold M, that is, of the indef- inite integrals the manifold supports, became a fundamental tool for understanding its global geometry. Many topological properties of a differentiable manifold M are encoded in its algebra of differential forms. An easily defined degree-1 derivation d on forms, generalizing implicit differentiation f → df on functions, turned the algebra of differential forms into a complex, called the deRham complex after the theory’s pioneer, Georges deRham. For a differential
2000 Mathematics Subject Classification. Primary 14C30. Reviewer partially supported by NSF grant DMS-0200895.