A characterization of surfaces whose universal cover is the bidisk

We show that the universal cover of a compact complex surface $X$ is the bidisk $\text{HH}\times \text{HH}$, or $X$ is biholomorphic to ${\text{PP}}^1\times {\text{PP}}^1$, if and only if $K_X^2>0$ and there exists an invertible sheaf $\eta$ such that ${\eta }^2\cong {\text{hol}}_X$ and $H^0(X,S^2{\mathrm{\Omega }}_X^1(-K_X)\otimes \eta )\mathrm{\ne }0$. 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.