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

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.

Motivic mirror symmetry for Higgs bundles

“…Since deg(D) > 2g − 2, this stability parameter lies in the cone ∆ Proof. The fact that this G m -action is semi-projective follows by the same argument for K C -Higgs bundles (see [39,Proposition 2.2]). As in [39, Proposition 2.5], one shows that semistability of the…”

“…In this section, we generalise our previous result concerning motives of moduli spaces of Higgs bundles [39,Theorem 1.1] to moduli spaces of D-twisted Higgs bundles. Throughout this section, we work over an arbitrary field k and assume that C(k) = ∅.…”

“…Hence, to finish the proof of Theorem 1.1, it remains to show that both source and target of ν D γ,mot are abelian. In order to do this, we extend our previous work [39] to prove that, for a divisor D either with D = K C or deg(D) > 2g − 2, the motive of the moduli space of D-twisted GL-Higgs bundles of coprime rank and degree is generated by the motive of the curve (and in particular is abelian). We then consider SL-Higgs bundles.…”

“…However, the cohomology of the SL-Higgs moduli space is not generated by tautological classes. On a motivic level, this difference between the case of moduli of vector bundles and Higgs bundles (with fixed determinant) is illustrated by the fact that the motives of the vector bundle moduli spaces N and N L are both generated by the motive of C (see [25,Proposition 4.1]), whereas in the case of Higgs bundles, although the motive of the GL-Higgs moduli space M is generated by the motive of C by [39], this is not true for the SL-Higgs moduli space M L for a general complex curve C by [26,Proposition 5.7].…”

“…Our second main result is the following theorem, which is proved in §3 and §4. The strategy for SL-Higgs bundles follows the same lines as for GL-Higgs bundles in [39] which was based on the geometric ideas in [29]: one uses a motivic Bia lynicki-Birula decomposition associated to the G m -action on the Higgs moduli space given by scaling the Higgs field to relate to motives of moduli spaces of chains of vector bundle homomorphisms (of fixed total determinant in the case of SL-Higgs bundles). From this, we deduce the purity of M (M D ) and M (M D L ) for Bondarko's weight structure and following [39, §6.3], it suffices to show that M (M D ) (resp.…”