TY - JOUR

T1 - A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space, II

AU - Kodama, Akio

AU - Shimizu, Satoru

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2006/7

Y1 - 2006/7

N2 - 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.

AB - 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.

KW - Holomorphic automorphism groups

KW - Holomorphic equivalences

KW - Torus actions

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U2 - 10.2969/jmsj/1156342031

DO - 10.2969/jmsj/1156342031

M3 - Article

AN - SCOPUS:33749676796

VL - 58

SP - 643

EP - 663

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 3

ER -