The C*-action and stratifications of the moduli space of semi-stable Higgs bundles of rank 5

Let X be a compact Riemann surface of genus g >= 2. The moduli space M(r, d) of rank r and degree d semi-stable Higgs bundles over X admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a C*-action on M(r, d) given by the product on the Higgs field, which provided the Bia & lstrok;ynicki-Birula stratification by considering the Hodge limit bundles limz -> 0(E, z <middle dot> phi). In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was r = 5, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank 5, the Shatz strata traversed the Bia & lstrok;ynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank 5 Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided.