Semistable principal Higgs bundles

TL;DR

This paper establishes a criterion for semistability of principal Higgs G-bundles on complex projective manifolds, linking semistability and Chern class conditions to the numerical effectiveness of associated line bundles.

Contribution

It provides a Miyaoka-type semistability criterion for principal Higgs bundles, connecting geometric stability conditions with numerical effectiveness of line bundles derived from group characters.

Findings

Semistability characterized by numerical effectiveness of line bundles.

Vanishing of second Chern class of the adjoint bundle linked to semistability.

Alternative characterizations via a new notion of numerical effectiveness for Higgs bundles.

Abstract

We give a Miyaoka-type semistability criterion for principal Higgs G-bundles E on complex projective manifolds of any dimension, i.e., we prove that E is semistable and the second Chern class of its adjoint bundle vanishes if and only if certain line bundles, obtained from the characters of the parabolic subgroups of G, are numerically effective. We also give alternative characterizations in terms of a notion of numerical effectiveness of Higgs vector bundles we have recently introduced.