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. Chapter 1 contains basic ideas, including the deformation of complex (analytic) structures on compact manifolds (without boundary), examples of the construction of complex manifolds and quadric transformations. In Chapter 2 cohomology with values in sheaves is defined and used to describe infinitesimal deformations, which are classes in the cohomology H1(M,Θ) of the manifold M with values in the sheaf Θ of holomorphic vector fields. If H1(M,Θ)=0, an elementary proof of the rigidity under local deformation of the complex structure of M is given. The cohomology sequence of a short exact sequence of sheaves of abelian groups is constructed, vector bundles (in particular, complex line bundles and their Chern classes) are discussed, and the chapter concludes with the Dolbeault resolution of the sheaf of sections of a holomorphic vector bundle. The third chapter is concerned with Kähler manifolds and vanishing theorems for cohomology with values in sheaves, and concludes with a proof of the Kodaira embedding theorem: Every Hodge manifold (compact complex manifold with a positive complex line bundle) is (projective) algebraic. In Chapter 4 the standard elliptic theory of partial differential equations is used to prove the existence of deformations: first the existence when there are no obstructions (more precisely, when H2(M,Θ)=0) and then existence in the general case (Kuranishi's theorem). Finally it is shown that all sufficiently small deformations of the complex structure of a (compact) Kähler manifold are Kähler manifolds (true only for small deformations as was shown by H. Hironaka [Ann. of Math. (2) 75 (1962), 190–208; MR0139182]).
Reviewer: Spencer, D. C. [form MathSciNet]