Stratifications and coarse moduli spaces for Higgs bundles via Non-Reductive GIT

The moduli space of semistable Higgs bundles on a smooth projective curve was first constructed by Nitsure in 1990, using Geometric Invariant Theory (GIT). It is a widely studied moduli space thanks to its rich geometric structure. The aim of this talk is to describe how recent results in Non-Reductive GIT can be used to construct new moduli spaces of Higgs bundles, including moduli spaces of unstable Higgs bundles. More precisely, I will describe two stratifications of the stack of Higgs bundles which satisfy the property that each stratum admits a quasi-projective coarse moduli space. The first is a refinement of the Higgs Harder-Narasimhan stratification (based on the instability type of a Higgs bundle), the second a refinement of the Harder-Narasimhan stratification (based on the instability type of the underlying bundle). I will explicitly describe these refined stratifications in the case of rank 2 Higgs bundles, and discuss the topology and geometry of the corresponding moduli spaces.