A characterization of surfaces whose universal cover is the bidisk

Abstract

We show that the universal cover of a compact complex surface $X$ is the bidisk $\HH \times \HH$, or $X$ is biholomorphic to $\PP^1 \times \PP^1$, if and only if $K_X^2>0$ and there exists an invertible sheaf $\eta$ such that $η^2\cong \hol_X$ and $H^0(X,S^2{\Omega }_X^1(-K_X)\otimes \eta )\ne 0$. The two cases are distinguished by the second plurigenus, $P_2(X)\ge 2$ in the former case, $P_2(X)=0$ in the latter. We also discuss related questions.

12 pages, references added