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

Lie Fu; Victoria Hoskins; Simon Pepin Lehalleur; Lie Fu; Victoria Hoskins; Simon Pepin Lehalleur

doi:10.3934/era.2022004

Electronic Research Archive

2022, Volume 30, Issue 1: 66-89. doi: 10.3934/era.2022004

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

Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

Received: 16 February 2021 Revised: 01 November 2021 Accepted: 22 November 2021 Published: 09 December 2021

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 Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.
- moduli spaces,
- vector bundles on curves,
- Higgs bundles,
- Chow motives,
- wall-crossing,
- flips and flops
Citation: Lie Fu, Victoria Hoskins, Simon Pepin Lehalleur. Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve[J]. Electronic Research Archive, 2022, 30(1): 66-89. doi: 10.3934/era.2022004

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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 Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.

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