A characterization of C^k x (C*)^ℓ from the viewpoint of biholomorphic automorphism groups

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