On Severi type inequalities

Abstract

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational invariants of a general fiber $F$ of the Albanese morphism of $X$. As applications, we provide a new lower bound of volumes of irregular threefolds, a sharp lower bound of varieties of maximal Albanese dimension and of general type, and show that the canonical model of such a variety with the minimal volume should be a flat double covers of a principally polarized abelian variety $(A, \Theta)$ branched over $D\in |2\Theta|$.