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

Abstract

In this paper, we prove that the holomorphic automorphism groups of the spaces ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$ and $({\mathbf{C}}^k-\{0\left\}\right)\times ({\mathbf{C}}^{*}{)}^{n-k}$ are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$: 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 ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$ as topological groups. Then $M$ itself is biholomorphically equivalent to ${\mathbf{C}}^k\times ({\mathbf{C}}^{*}{)}^{n-k}$. This was first proved by us in [5] under the stronger assumption that $M$ is a Stein manifold.