Generalized Clifford–Severi inequality and the volume of irregular varieties

Abstract

We give a sharp lower bound for the self-intersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the generalized Clifford–Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $vol\left(X\right)\ge 2n!\chi \left({\omega }_X\right)$ and it is sharp.