Generalized Clifford-Severi Inequality and the Volume of Irregular Varieties

TL;DR

This paper establishes a sharp lower bound for the self-intersection of nef line bundles on irregular varieties, extending to vector bundles and providing volume bounds, thereby advancing understanding of irregular varieties' geometry.

Contribution

It introduces the Generalized Clifford-Severi inequality, extending classical bounds to irregular varieties and nef vector bundles, with applications to volume estimates.

Findings

Lower bound for self-intersection in terms of global sections and Albanese dimension

Extension of bounds to nef vector bundles and slope inequalities

Sharp volume lower bound for maximal Albanese dimension varieties

Abstract

We give a sharp lower bound for the selfintersection 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 fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when $X$ is of maximal Albanese dimension the bound is $Vol(X) \geq 2 n! \chi\omega_X$ and it is sharp.