Pseudoconvexity in the theory of complex spaces

Abstract

The survey is devoted to various aspects of the theory of complex spaces related to the concept of pseudoconvexity, positive and negative vector bundles, and their generalizations.The present survey is devoted to various aspects of the theory of complex spaces connected with the concept of pseudoconvexity. The p-convex (or p-pseudoconvex) and q-concave spaces comprise a broad class of complex spaces which includes both Stein spaces as well as compact complex spaces; they were first introduced and studied in the work of Andreotti and Grauert [23] in 1962. The second chapter of the survey is devoted to the theory of these spaces which is itself a further development of the ideas related to the Levi problem. In the third chapter works are considered which are devoted to positive and negative vector bundles and their generalizations the study of which is strongly connected with the theory of pseudoconvexity. One of the applications of positive and negative bundles is criteria that a compact complex space be algebraic (Kodaira [195], Grauert [106]); here analytic geometry is closely coupled with algebraic geometry.In Chaps. II and IIIa survey is given of work on the topic indicated above which was reviewed in Referativnyi Zhurnal "Matematika" between January 1964 and January 1977; some later work is also considered. Chapter I is a direct continuation of the survey [8] devoted to Stein spaces and to which the entire present survey is closely related. In all cases unless otherwise specified we use the terminology of the paper [8]. In references to the bibliography of this paper the corresponding number is equipped with the letter S {e.g., [$29]).We note that our survey does not touch on a number of important areas of the theory of complex spaces related to the ideas of pseudoconvexity. These include the following questions: the theory of functions in pseudoconvex domains continuous up to the boundary or satisfying other boundary conditions (in this regard see the survey paper [11]) and also the eohomology theory of coherent analytic sheaves with estimates; various aspects of the theory of CR-manifolds; the differential calculus on complex spaces; duality theory. We also nearly fail to consider the theory of hulls of holomorphy and the theory of extension of analytic objects.We note several books on analytic geometry which have appeared since 1970. The book of Fischer [80] contains the foundations of the theory of complex spaces and their holomorphic mappings, in particular, the theory of Stein spaces, differential calculus, the theory of flat mappings, and the theory of meromorphic functions. Many theorems are presented there without proofs. Certain important areas of the theory which have been worked out in recent years are expounded in the book of Banica and Stanasila [36]: local eohomologies, finiteness theorems for proper and projective mappings, flat mappings, and the duality theory. The new edition of HSrmander's book [144] does not differ in any significant way from the previous edition. The books of Morrow a...