A remark on Frobenius descent for vector bundles

TL;DR

This paper explores vector bundles on relative curves over Spec Z, showing that some bundles have Frobenius descent in reductions but are not semistable in characteristic zero, and proves conditions for generic semistability.

Contribution

It provides examples of vector bundles with Frobenius descent in reductions that are not semistable in characteristic zero and establishes criteria for generic semistability on certain varieties.

Findings

Existence of vector bundles with Frobenius descent in reductions but not semistable in characteristic zero

Proof that vector bundles with Frobenius descent are generically semistable on certain varieties

Identification of classes of varieties where this property holds

Abstract

We give a class of examples of vector bundles on a relative smooth projective curve over Spec Z such that for infinitely many prime reductions the bundle has a Frobenius descent, but the restriction to the generic fiber in characteristic zero is not semistable. In the third section of the paper we prove for a large class of varieties (including abelian varieties) that any vector bundle with this Frobenius descent property is generically semistable.