A Remark on a Theorem by Kodama and Shimizu

Abstract

We prove a characterization theorem for the unit polydisc $\Delta^n\subset\CC^n$ 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 $\hbox{Aut}(M)$ of holomorphic automorphisms of $M$ acts on $M$ with compact isotropy subgroups, and (ii) $\hbox{Aut}(M)$ and $\hbox{Aut}(\Delta^n)$ are isomorphic as topological groups equipped with the compact-open topology, then $M$ is holomorphically equivalent to $\Delta^n$.