# On Severi type inequalities

## TL;DR

This paper investigates Severi type inequalities for big line bundles on irregular varieties, linking them to birational invariants of fibers and deriving bounds on volumes and structural properties of such varieties.

## Contribution

It introduces new Severi type inequalities for irregular varieties using cohomological rank functions and relates these to birational invariants of fibers, providing bounds and structural insights.

## Key findings

- New lower bounds for volumes of irregular threefolds
- Sharp bounds for varieties of maximal Albanese dimension and general type
- Canonical models are flat double covers of abelian varieties branched over divisors

## 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 $D\in |2\Theta|$.