  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 $D\in |2\Theta|$.

Algebraic Geometry