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

KODAMA, Akio ; SHIMIZU, Satoru

J. Math. Soc. Japan, Tome 58 (2006) no. 3, p. 643-663 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 2006-07-14
Classification:  holomorphic automorphism groups,  holomorphic equivalences,  torus actions,  32M05,  32Q28