Semistable and numerically effective principal (Higgs) bundles

TL;DR

This paper establishes criteria for semistability of principal Higgs bundles on complex projective manifolds using numerical effectiveness of associated line bundles, and introduces new notions of numerical flatness for principal bundles.

Contribution

It provides new semistability criteria based on line bundle positivity and introduces the concept of numerical flatness for principal (Higgs) bundles, extending existing theories.

Findings

Semistability after pullback to curves characterized by line bundle positivity.

Semistable Higgs bundles with vanishing second Chern class satisfy these criteria.

Numerically flat principal bundles admit reductions with flat Hermitian-Yang-Mills connections.

Abstract

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with real coefficients is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.