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J. Korean Math. Soc. 2003; 40(3): 563-575

Printed May 1, 2003

Copyright © The Korean Mathematical Society.

A characterization of $\bf C^k \times (\bf C^*)^{\ell}$ from the viewpoint of biholomorphic automorphism groups

Akio Kodama and Satoru Shimizu

Kanazawa University and Tohoku University

Abstract

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We show that if a connected Stein manifold $M$ of dimension $n$ has the holomorphic automorphism group $\text{Aut}(M)$ isomorphic to $\text{Aut} (\bf C^k \times (\bf C^*)^{n - k})$ as topological groups, then $M$ itself is biholomorphically equivalent to $\bf C^k \times (\bf C^*)^{n - k}$. Besides, a new approach to the study of $U(n)$-actions on complex manifolds of dimension $n$ is given.

Keywords: holomorphic automorphism groups, holomorphic equivalences, torus actions

MSC numbers: Primary 32M05; Secondary 32Q28

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