The main purpose of this book is to give an introduction to the Kodaira-Spencer theory of deformations of complex structures. The original proof of the Kodaira embedding theorem is given showing that the restricted class of Kähler manifolds called Hodge manifolds is algebraic. Included are the semicontinuity theorems and the local completeness theorem of Kuranishi. The book is based on notes taken by James Morrow from lectures given by Kunihiko Kodaira at Stanford University in 1965-1966. Complete references are given for the results that are used from elliptic partial differential equations. Readership Graduate students and research mathematicians interested in abstract complex manifolds. Reviews "This book, a revision and organization of lectures given by Kodaira at Stanford University in 1965-66, is an excellent, well-written introduction to the study of abstract complex (analytic) manifolds--a subject that began in the late 1940's and early 1950's. It is largely self-contained, except for some standard results about elliptic partial differential equations, for which complete references are given." *-- D. C. Spencer, MathSciNet* "The book under review is the faithful reprint of the original edition of one of the most influential textbooks in modern complex analysis and geometry. The classic "Complex Manifolds" by J. Morrow and K. Kodaira was first published in 1971 ..., essentially as a revised and elaborated version of a set of notes taken from lectures of Fields medallist Kunihiko Kodaira at Stanford University in 1965-1966, and has maintained its role as a standard introduction to the geometry of complex manifolds and their deformations ever since." *-- Werner Kleinert, Zentralblatt MATH* "Of course everyone knows Abel's exhortation that we should seek out "the masters, not their pupils," if we are to learn mathematics well and effectively. ... There is no question that this beautifully constructed book, full of elegant (and very economical) arguments underscores Abel's aforementioned dictum. Perhaps especially today, when so much is asked of the student of this material in the way of prerequisites, one can do no better than to turn to a master." *-- MAA Reviews* |