On motives of parabolic Higgs bundles and parabolic connections

Abstract

Let [Formula: see text] be a compact Riemann surface of genus [Formula: see text] and let [Formula: see text] be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over [Formula: see text] with the parabolic structure over [Formula: see text]. For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their [Formula: see text]-polynomials are the same. We also show that the Voevodsky and Chow motives of these two moduli spaces are also equal. We showed that the Grothendieck motivic classes and the [Formula: see text]-polynomials of parabolic Higgs moduli and of parabolic Hodge moduli are closely related. Finally, we considered the moduli spaces with fixed determinants and showed that the above results also hold for the fixed determinant case.