  Using the theory of cohomology support locus, we give a necessary condition
for the Albanese map of a smooth projective surface being a submersion. More
precisely, assuming 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.
