# Continuous CM-regularity of semihomogeneous vector bundles

## TL;DR

This paper establishes an upper bound on the Castelnuovo-Mumford regularity of M-regular sheaves on abelian varieties, with equality characterized by continuous global generation, providing insights into ample semihomogeneous vector bundles.

## Contribution

It proves a bound on the regularity of M-regular sheaves on abelian varieties and characterizes when equality occurs, advancing understanding of semihomogeneous vector bundles.

## Key findings

- Regularity is at most the dimension of the abelian variety.
- Equality holds when the dual bundle twisted by the line bundle is continuously globally generated.
- Provides a numerical criterion for ample semihomogeneous bundles.

## Abstract

We show that if $X$ is an abelian variety of dimension $g \geq 1$ and ${\mathcal E}$ is an M-regular coherent sheaf on $X$, the Castelnuovo-Mumford regularity of ${\mathcal E}$ with respect to an ample and globally generated line bundle ${\mathcal O}(1)$ on $X$ is at most $g$, and that equality is obtained when ${\mathcal E}^{\vee}(1)$ is continuously globally generated. As an application, we give a numerical characterization of ample semihomogeneous vector bundles for which this bound is attained.