On deep Frobenius descent and flat bundles

TL;DR

This paper explores conditions under which vector bundles on smooth projective families over arithmetic bases are semistable on the generic fiber, focusing on Frobenius descent data and arithmetic properties of closed fibers.

Contribution

It establishes new criteria linking Frobenius descent data on fibers to the semistability of vector bundles on the generic fiber in an arithmetic setting.

Findings

Frobenius descent data influences semistability of vector bundles

Arithmetic properties of fibers determine bundle behavior on the generic fiber

Conditions for semistability based on Frobenius pullbacks are identified

Abstract

Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a sequence (p_n, e_n)_{n \in \NN}, consisting of closed points p_n in Spec R and Frobenius descent data E_{p_n} \cong F^{e_n}^*(F) on the closed fibers X_{p_n}, the bundle E_0 on the generic fiber X_0 is semistable.