Stability of syzygy bundles on varieties of Picard number one

TL;DR

The paper proves a Hilbert-polynomial criterion for the slope stability of syzygy bundles $M_L$ associated to globally generated ample line bundles on smooth projective varieties of Picard number $1$, requiring $-K_X$ nef and a positivity condition on the Hilbert polynomial $P_H(t)$. Using vanishings of wedge powers and a monotonicity property of $P_H(t)$, it derives $\mu_H$-stability of $M_L$ and applies the criterion to a wide range of varieties, including complete intersections with nef $-K_X$, hyperkähler and abelian varieties, rational homogeneous spaces, and weak Calabi–Yau and Fano varieties of low dimension. For complete intersections, it further shows that $P_{\mathcal{O}_X(1)}(t)$ has non-negative coefficients when $\sum d_i\le n+1$, yielding stability of $M_{\mathcal{O}_X(1)}$; a similar strategy extends to many other Picard-number-one classes. In the hyperkähler case, a deformation-theoretic argument together with moduli of semistable sheaves and Rekuski’s wall results establishes stability for all hyperkähler varieties. Overall, the work broadens the regimes where syzygy bundles are known to be slope-stable and provides a unified technique via Hilbert polynomials and wedge-power vanishings.

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, 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 $\leq 4$, and Fano varieties of Picard number $1$ of dimension $\leq5$. Also we prove the stability of syzygy bundles on all hyperkähler varieties.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4: =Theorem \ref{['thm ci stable']}+Corollary \ref{['cor other varieties']}
• Theorem 1.5: see Theorem \ref{['thm ci poly>0']}
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2: coandua2011stability
• Lemma 2.3: green1984koszul, coandua2011stability