On varieties whose universal cover is a product of curves

TL;DR

This paper explores conditions under which a compact complex manifold's universal cover is a product of curves, focusing on the existence of a special tensor and its implications in low dimensions and Kähler surfaces.

Contribution

It introduces the concept of a semispecial tensor as a necessary condition and studies its sufficiency in characterizing manifolds with product of curves as universal covers.

Findings

Semispecial tensor is necessary for universal cover to be a product of curves.

In dimensions 2 and 3, the tensor condition is sufficient with additional hypotheses.

Characterization of Kähler surfaces with product of curves as universal cover.

Abstract

We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product $C^n$ of a curve $C = \PP^1 or \HH$: the existence of a semispecial tensor $\omega$. A semispecial tensor is a non zero section $ 0 \neq \omega \in H^0(X, S^n\Omega^1_X (-K_X) \otimes \eta) $), where $\eta$ is an invertible sheaf of 2-torsion (i.e., $\eta^2\cong \hol_X$). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension $n= 2$ or $ n= 3$; but it is not sufficient alone, even in dimension 2. In the case of K\"ahler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases.