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 are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least $$\frac{3}{8}$$ . We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least $$2(\dim X)!$$ . Moreover, if $${{\,\mathrm{vol}\,}}(X)=2(\dim X)!$$ , the canonical model of X is a double cover of a principally polarized abelian variety $$(A, \Theta )$$ branched over some divisor $$D\in |2\Theta |$$ .