On Severi Type Inequalities for Irregular Surfaces

Abstract

Let X be a minimal surface of general type and maximal Albanese dimension with irregularity q ≥ 2. We show that K2 X ≥ 4χ(OX) + 4(q − 2) if K2 X < 9 2 χ(OX), and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if K2 X ≥ 36(q−2) or χ(OX) ≥ 8(q−2), and we also prove a conjecture that the surfaces of general type and maximal Albanese dimension with K2 X = 4χ(OX) are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.