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Mixed Hodge structures on character varieties of nilpotent groups

Carlos Florentino, Sean Lawton, Jaime Silva

Abstract

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety R and on the character variety R//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.

Mixed Hodge structures on character varieties of nilpotent groups

Abstract

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety R and on the character variety R//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.

Paper Structure

This paper contains 17 sections, 26 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Character Varieties and Mixed Hodge Structures
  3. Character Varieties
  4. Mixed Hodge Structures
  5. Irreducible Components
  6. The Torus Component
  7. The Free Abelian Case
  8. Mixed Hodge Structure on Hom(F,G)
  9. Mixed Hodge Structures on a Smooth Model of $\mathcal{R}_{\mathbb{Z}^{r}}^{0}G$
  10. Mixed Hodge Structure on $\mathcal{R}_{\Gamma}^{0}G$ for Nilpotent $\Gamma$.
  11. Some computations for classical groups
  12. $G$-Equivariant Cohomology of $\mathcal{R}_{\Gamma}^{0}(G)$
  13. Mixed Hodge Structure on Hom(F,G)//G
  14. The MHS on $\mathcal{M}_{\Gamma}^{0}G$
  15. Examples for classical groups
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a finitely generated nilpotent group with abelian rank $r\geq1$, and $G$ a reductive $\mathbb{C}$-group. Then, both $\mathcal{M}_{\Gamma}^{0}G$ and $\mathcal{R}_{\Gamma}^{0}G$ are of Hodge-Tate type. More concretely, the MHS on $\mathcal{M}_{\Gamma}^{0}G$ coincides with the one of $T

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Example 3.1
  • Example 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • ...and 50 more