A characterization of surfaces whose universal cover is the bidisk

TL;DR

This paper characterizes complex surfaces with universal cover as the bidisk, identifying conditions involving the canonical bundle and sheaves that distinguish these surfaces from products of projective lines.

Contribution

It provides a precise criterion involving the canonical bundle and sheaf cohomology for when a surface's universal cover is the bidisk or a product of projective lines.

Findings

Universal cover is the bidisk if and only if specific cohomological conditions are met.

Surfaces with universal cover as  imes  are characterized by zero second plurigenus.

The paper distinguishes cases based on the second plurigenus P_2(X).

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 $\eta^2\cong \hol_X$ and $H^0(X, S^2\Omega^1_X (-K_X) \otimes \eta) \neq 0$. The two cases are distinguished by the second plurigenus, $P_2(X)\geq 2$ in the former case, $P_2(X)= 0$ in the latter. We also discuss related questions.