Cohomology groups with compact support for flat line bundles on certain complex Lie groups

Let X be a complex surface obtained as the quotient of the complex Euclidean space C² by a discrete subgroup of rank $3$. We investigate the cohomology group H₀¹(X, E) with compact support for a unitary flat line bundle E over X. We show the vanishing of H₀¹(X, E) for a certain class of such pairs $(X, E)$, which includes infinitely many examples such that H¹(X, E) is non-Hausdorff and infinite dimensional.