Stability of syzygy bundles on varieties of Picard number one

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, hyperkähler 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 $\le 4$, and Fano varieties of Picard number $1$ of dimension $\le 5$. Also we prove the stability of syzygy bundles on all hyperkähler varieties.