Function theory in several complex variables. Transl. from the Japanese by Norman Levenberg and Hiroshi Yamaguchi (Q2716073)

This book is an attempt to explain results in the theory of functions of several complex variables which were mostly established from the late 19th century through the middle of the 20th century. The focus is to introduce the mathematical world which was created by the author's advisor, Kiyoshi Oka (1901-1978). The author attempts to remain as close as possible to Oka's original work. NEWLINENEWLINENEWLINEKiyoshi Oka regarded the collection of problems in the study of domains of holomorphy as large mountains, and he believed that there could be no essential progress in analysis without climbing over these mountains. The work of Oka can be divided into two parts. The first is the study of analytic functions in univalent domains in \(\mathbb C^n\). Here he proved that three concepts: domains of holomorphy, holomorphically convex domains, and pseudoconvex domains, are equivalent; and, moreover, that the Poincaré, Cousin, and Runge problems -- when stated properly -- can be solved in domains of holomorphy satisfying the appropriate conditions. NEWLINENEWLINENEWLINEThe second part was to establish a method by which one can study analytic functions defined in a ramified domain over \(\mathbb C^n\) in which the branch points are considered as interior points of the domain. Oka's idea to treat analytic functions in a ramified domain has proved to be indispensable not only in analysis but also in other fields of mathematics. NEWLINENEWLINENEWLINEThis book consists of Parts I and II, according to Oka's work. In Part I the analytic functions are treated in a univalent domain in \(\mathbb C^n\), in Part II these are considered in an analytic space; this is a slight generalization of a ramified domain over \(\mathbb C^n\). The one exception to the adherence to Oka's program is the fact that a pseudoconvex univalent domain is a domain of holomorphy, proved in Part II in a more general setting by modifying Oka's original ideas. NEWLINENEWLINENEWLINEContents: Preface. Fundamental theory. Holomorphic functions and domains of holomorphy. Separate analyticity theorem. Implicit functions and analytic sets (local and global). Weierstrass condition. Projections of analytic sets in projective space. The Poincaré, Cousin, and Runge problems. Meromorphic functions. Pseudoconvex domains and pseudoconcave sets. Boundary problem. Analytic derived sets. Holomorphic mappings. Theory of analytic spaces. Ramified domains. Holomorphic functions on analytic sets. Universal denominators. \(\mathcal O\)-modules. Combination theorems. Local finiteness theorem. Analytic spaces. Analytic polyhedra. Stein spaces. Quantitative estimates. Representation of a Stein space. Normal pseudoconvex spaces. Linking problem. Principal theorem. Unramified domains over \(\mathbb C^n\). Bibliography. Index.