A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space, Ⅱ

摘要

In this paper, we prove that the holomorphic automorphism groups of the spaces C~k x (C~*)~(n-k) and (C~k - {0}) x (C~*)~(n-k) are not isomorphic as topo-logical groups. By making use of this fact, we establish the following characterization of the space C~k x (C~*)~(n-k): Let M be a connected complex manifold of dimension n that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that the holomorphic automorphism group of M is isomorphic to the holomorphic automorphism group of C~k x (C~*)~(n-k) as topological groups. Then M itself is biholomorphically equivalent to C~k x (C~*)~(n-k). This was first proved by us in [5] under the stronger assumption that M is a Stein manifold.