A Remark on a Theorem by Kodama and Shimizu

Abstract

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

6 pages