Stability of syzygy bundles on varieties of Picard number one

Abstract

We give a criterion for slope-stability of the syzygy bundle of a globally generated ample line bundle on a smooth projective variety of Picard number $1$ in terms of Hilbert polynomial. As applications, we prove the stability of syzygy bundles on many varieties, such as smooth Fano or Calabi--Yau complete intersections, hyperk\"ahler varieties of Picard number 1, abelian varieties of Picard number $1$, rational homogeneous varieties of Picard number 1, weak Calabi--Yau varieties of Picard number $1$ of dimension $\leq 4$, and Fano varieties of Picard number $1$ of dimension $\leq5$. Also we prove the stability of syzygy bundles on all hyperk\"ahler varieties.