On severi type inequalities for irregular surfaces

Let be a minimal surface of general type and of maximal Albanese dimension. We show that and we also obtain the characterization of the equality. As a consequence, we prove a conjecture that the surfaces of general type and of maximal Albanese dimension with are exactly the minimal resolution of the double covers of abelian surfaces branched over ample divisors with at worst simple singularities, and we also prove a conjecture of Manetti on the geography of irregular surfaces.