A remark on a theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ 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∈(Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .