Complex manifolds and deformation of complex structures. Transl. from the Japanese by Kazuo Akao
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.
scientific article; zbMATH DE number 2123715