Homogeneous principal bundles and stability

TL;DR

This paper investigates the stability properties of homogeneous principal bundles over rational homogeneous varieties, establishing conditions under which stability and equivariant stability coincide and exploring the existence of non-stable equivariant structures.

Contribution

It provides a detailed analysis of the relationship between stability and equivariant stability of homogeneous principal bundles, including new insights into their equivalence and examples of non-stable equivariant bundles.

Findings

Semistability and polystability are equivalent to equivariant versions.

Stable homogeneous principal bundles are always equivariantly stable.

Existence of non-stable equivariant structures on non-stable bundles.

Abstract

Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous principal H-bundle (E_H ,\rho) is equivariantly stable, but the converse is not true in general. If a homogeneous principal H-bundle (E_H ,\rho) is not equivariantly stable but not stable, then E_H admits an action \rho' of G such that the pair (E_H ,\rho') is a homogeneous principal H-bundle which is not equivariantly stable.