On the Severi type inequalities for irregular surfaces

Abstract

Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $q\ge 2$. We show that $K_X^2\ge 4\chi ({\mathcal{O}}_X)+4(q-2)$ if $K_X^2<\frac{9}{2}\chi ({\mathcal{O}}_X)$, and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if $K_X^2\ge 36(q-2)$ or $\chi ({\mathcal{O}}_X)\ge 8(q-2)$, and we also prove a conjecture that surfaces of general type and maximal Albanese dimension with $K_X^2=4\chi ({\mathcal{O}}_X)$ are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.

Any comment is welcome