Complex manifolds and deformation of complex structures. Transl. from the Japanese by Kazuo Akao
type: Item
instance of: scholarly article
title: Complex manifolds and deformation of complex structures. Transl. from the Japanese by Kazuo Akao
zbMATH Open document ID: 1058.32007
published in: Classics in Mathematics
publication date: 2004-12-22T00:00:00Z
review text: This book is a reprint of the beautiful classical book of Kodaira (1986) (see Zbl 0581.32012). It provides an introduction to the theory of deformations of complex structures on compact complex manifolds. The theory of deformations of complex manifolds is based on the theory of elliptic partial differential operators; an introduction to this is given in the appendix by Daisuke Fujiwara. The book is self-contained and structured as follows. In chapter 1 the basic tools on holomorphic functions in several variables are described. In chapter 2 are defined complex manifolds and examples are given: algebraic curves, algebraic surfaces, complex tori, Hopf manifolds. A surgery method to construct new complex manifolds from old ones is described. It ends with the notion of complex analytic families. Chapter 3 is dedicated to differential forms, vector bundles and sheaves. In chapter 4 is introduced the notion of infinitesimal deformation of complex structures, the main topic of the book. In chapter 5 is considered the existential problem. Each infinitesimal deformation of a complex manifold \(M\) induces an element of the first cohomology group \(H^{1}(M,\Theta)\), where \(\Theta\) is the sheaf of germs of holomorphic vector fields over \(M\). The existence problem asks, whether elements of \(H^{1}(M,\Theta)\) induced by are infinitesimal deformations. The number of parameters (moduli) of some examples are computed and observed that such a moduli number coincides with the dimension of \(H^{1}(M,\Theta)\). In chapters 6 and 7 are proved some of the statements given in the previous ones.
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32-01
Mathematics Subject Classification ID: 32G10
Mathematics Subject Classification ID: 58J10
zbMATH DE Number: 2123715
zbMATH Keywords: complex manifolds
zbMATH Keywords: deformation theory
zbMATH Keywords: moduli
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
author: Kunihiko Kodaira
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description: scientific article; zbMATH DE number 2123715