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Complex Manifolds and Deformation of Complex Structures

Kunihiko Kodaira, translated by Kazuo Akao
Springer Verlag
Publication Date: 
Number of Pages: 
Classics in Mathematics
SBN: 3-540-22614-1
[Reviewed by
Fernando Q. Gouvêa
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This is an unaltered paperback reprint, in Springer's Classics in Mathematics series, of Kodaira's classic book on complex manifolds, originally published in 1981 in Japanese and then translated into English in 1986. At the time, the "Telegraphic Review" described the book as "An introduction to the theory developed since the fifties (though with roots dating back to Riemann) by the author and D.C. Spencer. The first half is a general introduction to analysis and geometry on complex manifolds. A long appendix treats elliptic partial differential operators on manifolds."

Kodaira's work with Spencer was the starting point of a theory that continues to be active, with its own AMS subject classification (32Gxx). This classic exposition by one of the authors tries to capture the excitement of the actual discovery. In the introduction, Kodaira says that "The process of the development [of the theory of deformation on compact complex manifolds] was the most interesting experience in my whole mathematical life. It was similar to an experimental science developed by the interaction between experiments (examination of examples) and theory. In this book I have tried to reproduce this interesting experience; however I could not fully convey it. Such an experience may be a passing phenomenon which cannot be reproduced." However unsuccessful, it was an attempt worth making, and the result is a book worth having.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME.

Holomorphic Functions.- Complex Manifolds.- Differential Forms, Vector Bundles, Sheaves.- Infinitesimal Deformation.- Theorem of Existence.- Theorem of Completeness.- Theorem of Stability.- Appendix: Elliptic Partial Differential Operators on a Manifold by Daisuke Fujiwara.- Bibliography.- Index.