On Severi Type Inequalities for Irregular Surfaces

Let X be a minimal surface of general type and of maximal Albanese dimension. We show that K-X(2) >= 4 chi(O-X) + 4(q - 2), if K-X(2) < 9/2 chi(O-X) 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 K-X(2) = 4 chi(O-X) 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 if K-X(2) >= 36(q - 2) or chi(O-X) >= 8(q - 2).]