Motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a curve

TL;DR

This paper derives formulas for the rational Chow motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a smooth projective curve, using motivic decompositions and wall-crossing techniques.

Contribution

It introduces criteria to lift identities in the Grothendieck group to isomorphisms in the Chow motives category, advancing the understanding of moduli space motives.

Findings

Formulas for Chow motives of rank 3 vector bundle moduli spaces

Motivic Bialynicki-Birula decompositions for Higgs moduli spaces

Application of wall-crossing in motive calculations

Abstract

We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action.