On the motives of moduli of parabolic chains and parabolic Higgs bundles

TL;DR

This paper develops an algorithm to compute the virtual motives of moduli spaces of parabolic Higgs bundles on punctured surfaces, advancing the understanding of their geometric and topological properties.

Contribution

It introduces a localization-based algorithm for calculating the motives of moduli spaces of parabolic Higgs bundles with fixed invariants.

Findings

Algorithm successfully computes motives in the Grothendieck ring.

Provides new insights into the structure of moduli spaces of parabolic Higgs bundles.

Enhances tools for studying the topology of moduli spaces in algebraic geometry.

Abstract

Like the Higgs bundles on a Riemann surface who played an important role in the study of representation of the fundamental group of the surface, the parabolic Higgs bundles play also their importance in the study of the fundamental group but of punctured surface. In this paper, we shall give an algorithm to calculate the (virtual) motive (i.e in a suitable Grothendieck ring) of the moduli spaces of parabolic Higgs bundles of fixed rank, fixed degree and fixed parabolic structure, using localization with respect to the circle action.