On the vanishing of weight one Koszul cohomology of abelian varieties

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 (Np) (G. Pareschi, J. Amer. Math. Soc. 13 (2000) 651-664).