—Holomorphic mappings of complex spaces

Abstract

In ?3, we consider the existence of holomorphic maps with a given rank on complex spaces countable at infinity. We use the method given here to make a remark on an imbedding theorem for holomorphically complete spaces due to R. Remmert [4]. The author is indebted to Professor H. Cartan for his suggestions and for pointing out that the space of holomorphic functions is complete even for non-normal spaces. The author's thanks are due also to Professor K. Chandrasekharan for his encouragement prior to and during the preparation of this note. 2. We shall denote, throughout this note, the space of holomorphic functions on a complex space X, with the compact convergence topology, by R(X) =R. If X is holomorphically separable, it can be shown that X is Kcomplete (in the sense of [1 ]; we note that it is enough to have, for any xoCX, a holomorphic map f: X-->Ck such that xo is an isolated point of f'f(xo)), so that, by [1, Satz 8], X is countable at infinity. Hence R(X) is metrisable, and by [2, Satz 28], it is complete. We prove first