Rationality of the instability parabolic and related results

TL;DR

This paper investigates the stability properties of principal bundles under Frobenius pullbacks in positive characteristic, establishing conditions under which semistability is preserved when extending structure groups via a representation.

Contribution

It proves the existence of a finite number of Frobenius pullbacks ensuring semistability transfer through structure group extension for principal bundles.

Findings

Existence of an integer t for Frobenius pullbacks preserving semistability.

Quantification of the number of Frobenius pullbacks needed.

Extension of semistability results to principal G-bundles in positive characteristic.

Abstract

In this paper we study the extension of structure group of principal bundles with a reductive algebraic group as structure group on smooth projective varieties defined over algebraically closed field of positive characteristic. Our main result is to show that given a representation {\rho} of a reductive algebraic group G, there exists an integer t such that any semistable G-bundle whose first t frobenius pullbacks are semistable induces a semistable vector bundle on extension of structure group via {\rho}. Moreover we quantify the number of such frobenius pullbacks required.