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

Abstract

Let $ X $ be a compact Riemann surface of genus $ g\geq 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}^* $-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 $ \lim_{z\to 0}(E, z\cdot\varphi) $. 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.

Keywords

• higgs bundle
• stratification
• action
• harder-narasimhan type
• semi-stable
• moduli space