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On motives of parabolic Higgs bundles and parabolic connections

Sumit Roy

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.

On motives of parabolic Higgs bundles and parabolic connections

Abstract

Let be a compact Riemann surface of genus and let be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over with the parabolic structure over . For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their -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 -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.
Paper Structure (22 sections, 18 theorems, 93 equations)

This paper contains 22 sections, 18 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Future works
  3. Preliminaries
  4. Parabolic bundles
  5. Parabolic Higgs bundles
  6. Hitchin fibration
  7. Parabolic connections
  8. Parabolic $\lambda$-connections
  9. Semiprojectivity of the moduli spaces
  10. Semiprojectivity of the moduli space of parabolic Higgs bundles
  11. Semiprojectivity of parabolic Hodge moduli space
  12. Grothendieck motives of semiprojective varieties
  13. Grothendieck ring of varieties
  14. Mixed Hodge structure and $E$-polynommial
  15. Voevodsky motives and Chow motives
  16. ...and 7 more sections

Key Result

Theorem 1.1

In $\hat{K}(\mathcal{V}_\mathbb{C})$ we have the following motivic equalities Therefore, we have the following equalities of the $E$-polynomials

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark
  • Definition 2.6
  • Definition 2.7
  • ...and 41 more