On the vanishing of weight one Koszul cohomology of abelian varieties

TL;DR

This paper proves the vanishing of certain twisted Koszul cohomology groups for abelian varieties with ample line bundles, extending understanding of their algebraic and geometric properties.

Contribution

It establishes new vanishing results for Koszul cohomology groups, complementing previous work on property $(N_p)$ for abelian varieties.

Findings

Vanishing of $K_{p,1}$ groups for abelian varieties with ample line bundles

Extension of property $(N_p)$ results

Provides tools for understanding syzygies of abelian varieties

Abstract

In this Note we prove the vanishing of (twisted) Koszul cohomology groups $K_{p,1}$ of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property $(N_p)$.