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 $D\in |2\Theta |$.

arXiv commentcorrect some typos and add some references

References in corpus (7)

• The Severi inequality $K^2\ge 4\chi$ for surfaces of maximal Albanese dimension
• Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
• Generalized Clifford-Severi Inequality and the Volume of Irregular Varieties
• Explicit birational geometry of 3-folds and 4-folds of general type, III
• Some results on the eventual paracanonical maps
• Relative Clifford inequality for varieties fibered by curves
• Cohomological rank functions on abelian varieties