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

Abstract

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