On the Voevodsky motive of the moduli space of Higgs bundles on a curve

TL;DR

This paper investigates the motive of the moduli space of semistable Higgs bundles on a curve, establishing its relation to the curve's motive and proving a motivic non-abelian Hodge correspondence in characteristic zero.

Contribution

It demonstrates that the motive of the Higgs moduli space is generated by the curve's motive and establishes an isomorphism of motives between Higgs and de Rham moduli spaces in characteristic zero.

Findings

The motive of the Higgs moduli space lies in the subcategory generated by the curve's motive.

In characteristic zero, the motives of Higgs and de Rham moduli spaces are isomorphic.

The result connects the geometry of Higgs bundles with Voevodsky's motives.

Abstract

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.