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Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

David Alfaya, André Oliveira

Abstract

Let $\mathcal{L}=(L,[\cdot\,,\cdot],δ)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove that the motivic class of the moduli space of $\mathcal{L}$-connections of rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroid structure $[\cdot\,,\cdot]$ and $δ$ of $\mathcal{L}$ and neither on the line bundle $L$ itself, but only on the degree of $L$ (and of course on $r$, $d$ and $X$). In particular it is equal to the motivic class of the moduli space of $K_X(D)$-twisted Higgs bundles of rank $r$ and degree $d$, for $D$ any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding $E$-polynomials. Some applications of these results are then deduced.

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

Abstract

Let be an algebraic Lie algebroid over a smooth projective curve of genus such that is a line bundle whose degree is less than . Let and be coprime numbers. We prove that the motivic class of the moduli space of -connections of rank and degree over does not depend on the Lie algebroid structure and of and neither on the line bundle itself, but only on the degree of (and of course on , and ). In particular it is equal to the motivic class of the moduli space of -twisted Higgs bundles of rank and degree , for any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding -polynomials. Some applications of these results are then deduced.

Paper Structure

This paper contains 37 sections, 56 theorems, 316 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On the moduli spaces of Lie algebroid connections
  3. The classical case: moduli spaces of flat connections and of Higgs bundles
  4. This paper's contribution
  5. Organization
  6. Open questions and future work
  7. Moduli spaces of twisted Higgs bundles
  8. Lie algebroid connections, $\Lambda$-modules and moduli spaces
  9. Lie algebroids and ${\mathcal{L}}$-connections
  10. $\Lambda$-modules and ${\mathcal{L}}$-connections
  11. Moduli spaces of ${\mathcal{L}}$-connections and of $\Lambda$-modules
  12. The ${\mathcal{L}}$-Hodge moduli spaces
  13. Classification of Lie algebroid structures on line bundles
  14. Smoothness and semiprojectivity of ${\mathcal{L}}$-Hodge moduli spaces
  15. Semiprojectivity of the moduli space of Higgs bundles
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth projective complex curve of genus $g\geq 2$. Let ${\mathcal{L}}=(L,[\cdot\,,\cdot],\delta)$ be an algebraic Lie algebroid on $X$ such that $L$ is a line bundle of degree $\deg(L)<2-2g$. Suppose $r,d$ are coprime integers. Then ${\mathcal{M}}_{{\mathcal{L}}}(r,d)$ is a non-empty,

Figures (1)

  • Figure 1: For any pair of line bundles $L$ and $L'$ of the same degree $d_L$, all moduli spaces of Lie algebroid connections parameterized by both spaces $H^0(L^{-1}\otimes T_X)$ and $H^0((L')^{-1}\otimes T_X)$ share the same motives (and other properties).

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • ...and 114 more