Higgs reductions and numerically flat principal Higgs bundles

TL;DR

This paper studies principal Higgs bundles with a specific flatness condition, proving they are either stable or reducible to stable, flat components, and explores their cohomological properties and conditions for Hermitian flatness.

Contribution

It establishes a criterion for H-nflat principal Higgs bundles to be stable or reducible to stable, flat components, and links their cohomology to associated graded objects.

Findings

H-nflat principal Higgs bundles are either stable or reducible to stable, flat components.

Cohomology of H-nflat bundles is isomorphic to that of their associated graded objects.

Vanishing second Chern class implies Hermitian flatness and trivial cohomology.

Abstract

I consider principal Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced by Bruzzo and Gra\~na Otero. I prove that a principal Higgs bundle $\mathfrak{E}=(E,\varphi)$ is H-nflat is either stable or there exists a Higgs reduction of $\mathfrak{E}$ to a parabolic subgroup $P$ of $G$ such that the principal $L$-bundle $\mathfrak{E}_L$ obtained by extending the reduced Higgs bundle $\mathfrak{E}_P$ to the Levi factor $L$ is H-nflat and stable; and as consequence, $H^{*}(\mathfrak{E},\mathbb{R})$ is isomorphic to the cohomology ring of the associated graded object $\mathrm{Gr}(\mathfrak{E})$ with coefficients in $\mathbb{R}$. Moreover, if $c_2(\mathrm{Ad}(E))$ vanishes then $\mathfrak{E}_L$ is also Hermitian flat and $H^{*}(\mathrm{Gr}(\mathfrak{E}),\mathbb{R})$ is trivial.