The ${\mathbb{C}}^{\ast }$-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $5$

Let $X$ be a compact Riemann surface of genus $g\ge 2$. The moduli space $\mathcal{M}(r,d)$ 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 ${\mathbb{C}}^{\ast }$-action on $\mathcal{M}(r,d)$ given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles $\underset{z\to 0}{lim}(E,z\cdot \phi )$. 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 Białynicki-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.