Exterior power of stable vector bundle destabilized by Frobenius pull-back

TL;DR

This work shows that in positive characteristic, the exterior power operation can destroy stability: on curves with genus $g\ge2$, there exist stable bundles whose exterior square after Frobenius pushforward, $F^n_*E\wedge F^n_*E$, is not semi-stable under broad conditions ($r>1$, $p>3$, or $n>1$). The authors develop a framework based on a canonical filtration of $F^*(F_*E)$ and slope computations to establish injections into wedge bundles and compare their slopes, leading to non-semistability results. They also demonstrate the existence of stable bundles that are not cohomologically stable, and discuss a conjecture linking (semi-)stability with cohomological semi-stability in positive characteristic. Overall, the paper deepens the understanding of how Frobenius and exterior operations interact with stability on curves in characteristic $p>0$ and provides concrete counterexamples to stability-preserving properties.

Abstract

In this paper, we prove that for any smooth projective curve $C$ of genus $g\geq2$ over an algebraically closed field of positive characteristic, there exists a stable vector bundle over $C$ whose exterior power is not semi-stable.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Conjecture 1.4
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5