A Remark on a Theorem by Kodama and Shimizu

Abstract

We prove a characterization theorem for the unit polydisc Δⁿ ⊂ ℂⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δⁿ.