A Remark on a Theorem by Kodama and Shimizu

TL;DR

This paper characterizes the unit polydisc in complex n-space by examining the automorphism group of a complex manifold, showing that certain symmetry conditions imply the manifold is equivalent to the polydisc.

Contribution

It provides a new characterization theorem for the polydisc based on automorphism group properties, extending previous results by Kodama and Shimizu.

Findings

Automorphism group acts with compact isotropy subgroups.

Isomorphism of automorphism groups implies manifold is the polydisc.

The result generalizes known characterizations of the polydisc.

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 $\Delta^n$.