An overview on curve semistable and numerically flat Higgs bundles
Armando Capasso
TL;DR
The paper extends Miyaoka's curve semistability and positivity notions to Higgs bundles via Higgs-Grassmann schemes, defining Higgs-ample, Higgs-nef, and Higgs-numerically flat objects with a recursive quotient criterion. It proves that curve semistability for Higgs bundles is equivalent to nefness of the Higgs-derived classes $\theta_t(\mathfrak{E})$ and $\lambda_t(\mathfrak{E})$, and that curve semistability implies semistability with respect to all polarizations, along with vanishing discriminants in the vector-bundle case. The work also presents a discriminant conjecture $\Delta(E)=0$ for curve semistable Higgs bundles and discusses cases where this holds, as well as the open problems surrounding the Chern classes of Higgs-nflat bundles. By connecting Higgs positivity, curve semistability, and Chern-class vanishing, the paper lays groundwork for a Higgs-variant of Miyaoka-type criteria and suggests directions for resolving the conjectures in higher rank and general backgrounds.
Abstract
After recalling the basic notions concerning Higgs-Grassmannian schemes, I review how these latter can be used to define generalisations of the notion of positivity conditions, such as numerically flatness, which "feel" the Higgs field. Then I prove several properties of Higgs bundles, over smooth projective varieties defined over an algebraically closed field of characteristic $0$, satisfying these conditions. Finally, I discuss how one can relate them to semistability of the so-called "curve semistable" Higgs bundles.