Motivic invariants of moduli spaces of rank 2 Bradlow - Higgs triples

TL;DR

This thesis investigates the geometry and invariants of moduli spaces of Bradlow-Higgs triples on curves, analyzing their behavior under stability parameter changes and establishing formulas relating their cohomology to known structures.

Contribution

It introduces a detailed study of the wall-crossing phenomena for rank 2 Bradlow-Higgs triples and derives a cohomological formula connecting these moduli spaces to Higgs bundle invariants.

Findings

Analysis of moduli space changes as stability parameter varies

Derivation of a cohomology formula relating small sigma moduli to Higgs bundle cohomology

Partial generalization to higher rank cases

Abstract

In the present thesis we study the geometry of the moduli spaces of Bradlow-Higgs triples on a smooth projective curve $C$. $(E,\varphi, s)$ is a Bradlow-Higgs triple if $(E,\varphi)$ is a Higgs bundle and $s$ is a non-zero global section of $E$. There is a family of stability conditions for triples that depends on a positive real parameter $\sigma$. The moduli spaces $\mathcal{M}_\sigma^{r,d}$ of $\sigma$-semistable triples of rank $r$ and degree $d$ vary with $\sigma$. The phenomenon arising from this is known as wall-crossing. In the first half of the thesis we will examine how the moduli spaces $\mathcal{M}_\sigma^{r,d}$ and their universal additive invariants change as $\sigma$ varies, for the case $r=2$. In particular we will study the case of $\sigma$ very close to 0, for which $\mathcal{M}_\sigma^{r,d}$ relates to the moduli space of stable Higgs bundles, and $\sigma$ very large, for which $\mathcal{M}_\sigma^{r,d}$ is a relative Hilbert scheme of points for the family of spectral curves. Some of these results will be generalized to Bradlow-Higgs triples with poles. In the second half we will prove a formula relating the cohomology of $\mathcal{M}_\sigma^{2,d}$ for small $\sigma$ and $d$ odd and the perverse filtration on the cohomology of the moduli space of stable Higgs bundles. The formula is not far from the generalized Macdonald formulas found in the works of Migliorini-Shende, Maulik-Yun and Migliorini-Shende-Viviani. We will also partially generalize this result to the case of rank greater than 2.