Frobenius pull backs of vector bundles in higher dimensions

TL;DR

This paper investigates how Frobenius pull-backs affect the Harder-Narasimhan filtration of vector bundles over higher-dimensional varieties, establishing conditions under which the filtration is refined and providing bounds on instability.

Contribution

It generalizes the behavior of Frobenius pull-backs of vector bundles to higher dimensions and principal bundles, with explicit bounds and necessary conditions on the characteristic p.

Findings

Harder-Narasimhan filtration of Frobenius pull-back refines the original under certain p bounds.

Provides bounds on the instability degree of Frobenius pull-backs.

Examples show the necessity of lower bounds on p.

Abstract

Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the Harder-Narasimhan filtration of $E$, provided there is a lower bound on the characteristic $p$ (in terms of rank of $E$ and the slope of the destabilising sheaf of the cotangent bundle of $X$). We also recall some examples, due to Raynaud and Monsky,to show that some lower bound on $p$ is necessary. We further prove an analogue of this result for principal $G$-bundles over $X$. We also give a bound on the instability degree of the Frobenius pull back of $E$ in terms of the instability degree of $E$ and well defined invariants ot $X$ and $E$.