Yang-Mills-Higgs connections on Calabi-Yau manifolds, II

Abstract

In this paper we study Higgs and co-Higgs $G$-bundles on compact Kähler manifolds $X$. Our main results are: (1) If $X$ is Calabi-Yau, and $(E,\theta )$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the principal $G$-bundle $E$ is semistable. In particular, there is a deformation retract of ${\mathcal{M}}_H(G)$ onto $\mathcal{M}(G)$, where $\mathcal{M}(G)$ is the moduli space of semistable principal $G$-bundles with vanishing rational Chern classes on $X$, and analogously, ${\mathcal{M}}_H(G)$ is the moduli space of semistable principal Higgs $G$-bundles with vanishing rational Chern classes. (2) Calabi-Yau manifolds are characterized as those compact Kähler manifolds whose tangent bundle is semistable for every Kähler class, and have the following property: if $(E,\theta )$ is a semistable Higgs or co-Higgs vector bundle, then $E$ is semistable.

12 pages. v2: references added. v3: final version, to be published in the Proceedings of GEOQUANT 2015, Travaux en Course, Université du Luxembourg