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

In this paper, we prove that the holomorphic automorphism groups of the spaces Ck × (C*) n-k and (Ck - 0) × (C*) n-k are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space Ck × (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 Ck × (C*) n-k as topological groups. Then M itself is biholomorphically equivalent to Ck × (C*) n-k. This was first proved by us in 5 under the stronger assumption that M is a Stein manifold. 全文公開200907