On motives of parabolic Higgs bundles and parabolic connections

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.