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 $(N_p)$ (G. Pareschi, J. Amer. Math. Soc. 13 (2000) 651–664).