Stability of syzygy bundles on certain Picard rank one varieties

Technical Summary: Stability of Syzygy Bundles on Certain Picard Rank One Varieties

Problem Statement
The paper addresses the stability of syzygy bundles $M_L$ on smooth projective varieties $X$ of dimension at least 2 with Picard rank 1. Given a nontrivial globally generated line bundle $L$ on $X$, the syzygy bundle $M_L$ is defined as the kernel of the evaluation map $H^0(X, L) \otimes \mathcal{O}_X \to L$. The central question (Question 1.1) is whether $M_L$ is stable with respect to the unique polarization on $X$ (since Picard rank is 1). While stability is well-understood for curves and known for specific higher-dimensional cases (e.g., Fano or Calabi-Yau varieties, or when $L$ is "sufficiently ample"), the general case for arbitrary ample $L$ on Picard rank 1 varieties remained open.

Methodology
The author employs a combination of algebraic geometry techniques, specifically focusing on the monotonicity of global sections of line bundles and slope stability criteria.

1. Monotonicity of Global Sections: A core technical contribution is the proof of a monotonicity result (Proposition 3.2 and Corollary 3.5). The author establishes that for a normal projective variety $X$ of dimension $\ge 2$, if $H$ is an ample globally generated line bundle and $F$ is a numerically trivial line bundle, the function $h^0(X, F(k))/k$ is non-decreasing for $k \ge 1$.

   • This is derived using the theory of graded rings and modules. By constructing a graded ring $R = \bigoplus H^0(X, \mathcal{O}_X(k))$ and a module $M = \bigoplus H^0(X, F(k))$, the author utilizes a regular sequence of general sections to relate the Hilbert polynomials of $M$ and its quotient.
   • The proof relies on the fact that for $k \ge 1$, the difference in the normalized dimensions of global sections is non-negative, leveraging the properties of numerically trivial bundles on normal varieties.

2. Contradiction via Slope Inequalities: To prove the main theorem, the author assumes $M_L$ is unstable. By standard results (citing [10]), instability implies the existence of a destabilizing subsheaf, specifically a line bundle $N$ and an integer $r$ such that the slope $\mu_H(\wedge^r M_L \otimes N) \le 0$ and $H^0(X, \wedge^r M_L \otimes N) \neq 0$.

   • The author expresses $L$ and $N$ in terms of a generator of the Néron–Severi group and numerically trivial bundles: $L \cong N_2(l)$ and $N \cong N_1(k)$ with $N_1, N_2$ numerically trivial.
   • Using the slope formula for syzygy bundles, the instability condition yields an inequality relating $k$, $l$, and $r$.
   • The monotonicity result (Corollary 3.5) is then applied to compare the growth of global sections $h^0(X, N)$ and $h^0(X, L)$. This leads to a chain of inequalities that forces $k \ge l$, which directly contradicts the earlier deduction that $k < l$ derived from the rank constraints.

Key Contributions and Results

• Theorem A: The primary result states that if $X$ is a smooth projective variety of dimension $\ge 2$ with Picard rank 1, and every ample line bundle on $X$ is globally generated, then for every ample line bundle $L$, the syzygy bundle $M_L$ is stable.
• Corollary 4.1: The theorem is applied to smooth complete intersections of dimension $\ge 2$ in projective space $\mathbb{P}^N$. By the Lefschetz theorems, these varieties have Picard rank 1 and generated by the hyperplane class (which is globally generated). Thus, the syzygy bundle of any nontrivial globally generated line bundle on such a variety is stable.
• Extension of Previous Work: The paper notes that while stability was previously known for Fano or Calabi-Yau complete intersections (via [10]), this result extends the affirmative answer to all smooth complete intersections of dimension $\ge 2$ with Picard rank 1, regardless of whether they are Fano, Calabi-Yau, or of general type.

Significance and Claims
The paper claims to provide an affirmative answer to Question 1.1 for a "large class of varieties," specifically those where all ample line bundles are globally generated. The author explicitly states that this extends earlier results by Jiang-Ren, Coandă, and others.

The significance is framed as resolving a natural question regarding the stability of syzygy bundles in the absence of "sufficient ampleness" assumptions on $L$. The paper does not claim to solve Question 1.1 for all Picard rank 1 varieties (noting that the hypothesis of global generation for all ample bundles is required), but rather establishes a robust class of examples (complete intersections) where the conjecture holds. The author also references Remark 4.2, noting that via Mehta-Ramanathan restriction theorems, the general question reduces to the surface case, and this work provides affirmative answers for a large class of surfaces of general type.