On motives of parabolic Higgs bundles and parabolic connections

TL;DR

This paper demonstrates that the moduli spaces of parabolic Higgs bundles and parabolic connections over a Riemann surface share identical motivic classes, $E$-polynomials, and motives, revealing deep geometric and motivic equivalences.

Contribution

It establishes the equality of motivic classes, $E$-polynomials, and motives for these moduli spaces, including fixed determinant cases, for generic weights.

Findings

Equal Grothendieck motivic classes for the two moduli spaces

Identical $E$-polynomials for the moduli spaces

Voevodsky and Chow motives of the spaces are equal

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 2$ and let $D\subset X$ be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over $X$ with the parabolic structure over $D$. For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their $E$-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 $E$-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.