An overview on curve semistable and numerically flat Higgs bundles

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.