Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface to be a submersion. More precisely, assuming that the cohomology support locus of any finite abelian cover of a smooth projective surface consists of finitely many points, we prove that the surface has trivial first Betti number, or is a ruled surface of genus one, or is an abelian surface.
"Algebraic Surfaces with Zero-dimensional Cohomology Support Locus." Taiwanese J. Math. 22 (3) 607 - 614, June, 2018. https://doi.org/10.11650/tjm/170808