Stratifications on the Moduli Space of Higgs Bundles

Abstract

AbstractIn this final chapter we review certain results on stratifications of the moduli space of Higgs bundles, performed with the invariants provided by the Harder-Narasimhan filtrations. The moduli space of Higgs bundles has two stratifications. The Shatz stratification arises from the Harder-Narasimhan type of the underlying vector bundle of the Higgs bundle, and the Białynicki-Birula stratification comes from the action of the nonzero complex numbers by multiplication on the Higgs field. While these two stratifications coincide in the case of rank two Higgs bundles, this is not the case in higher rank. We thus analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles, based on results contained in [31, 98,99,100]. This relation allows some applications, as the computation of homotopy groups of the moduli space.