Stability of syzygy bundles on varieties of Picard number one

TL;DR

This paper establishes a criterion for the slope-stability of syzygy bundles on smooth projective varieties with Picard number one, and demonstrates stability across various classes of such varieties including Fano, Calabi–Yau, and hyperkähler types.

Contribution

It provides a new criterion for slope-stability of syzygy bundles based on Hilbert polynomial and proves stability for numerous classes of varieties with Picard number one.

Findings

Proves stability of syzygy bundles on Fano and Calabi–Yau complete intersections

Establishes stability on hyperkähler and abelian varieties of Picard number 1

Demonstrates stability on rational homogeneous and weak Calabi–Yau varieties of dimension ≤4

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.