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Triangular decomposition of character varieties

Julien Korinman

TL;DR

This work extends character varieties to marked surfaces via stated character varieties, yielding affine Poisson spaces that behave well under gluing and triangulation. It builds a representation- and groupoid-based framework, introduces discrete models, and proves that curve functions generate the coordinate rings, enabling a triangular decomposition of the moduli space. The paper then constructs twisted groupoid (co)homologies to identify tangent and cotangent spaces with twisted cohomology and defines a canonical intersection form driving a generalized Goldman bracket that is triangulation-independent. In the abelian case it connects to relative cohomology and Chekhov–Fock-type structures, and it culminates with fusion (Alekseev–Malkin) results showing the stated character varieties are twisted quasi-Poisson or classical Poisson moduli spaces, coherently linking to skein algebras and quantum moduli spaces.

Abstract

A marked surface is a compact oriented surface equipped with some pairwise disjoint arcs embedded in its boundary. In this paper, we extend the notion of character varieties to marked surfaces, in such a way that they have a nice behaviour for the operation of gluing two boundary arcs together. These stated character varieties are affine Poisson varieties which coincide with the Culler-Shalen character varieties when the surface is unmarked and are closely related to the Fock-Rosly and Alekseev-Kosmann-Malkin-Meinrenken constructions in the marked case. These Poisson varieties are the classical moduli spaces underlying stated skein algebras and share similar properties. In particular, stated character varieties admit triangular decompositions, associated to triangulations of the surface. We identify the Zariski tangent spaces of these varieties with some twisted groupoid cohomological groups and provide a generalization of Goldman's formula for the Poisson bracket of curve functions in terms of intersection form in homology.

Triangular decomposition of character varieties

TL;DR

This work extends character varieties to marked surfaces via stated character varieties, yielding affine Poisson spaces that behave well under gluing and triangulation. It builds a representation- and groupoid-based framework, introduces discrete models, and proves that curve functions generate the coordinate rings, enabling a triangular decomposition of the moduli space. The paper then constructs twisted groupoid (co)homologies to identify tangent and cotangent spaces with twisted cohomology and defines a canonical intersection form driving a generalized Goldman bracket that is triangulation-independent. In the abelian case it connects to relative cohomology and Chekhov–Fock-type structures, and it culminates with fusion (Alekseev–Malkin) results showing the stated character varieties are twisted quasi-Poisson or classical Poisson moduli spaces, coherently linking to skein algebras and quantum moduli spaces.

Abstract

A marked surface is a compact oriented surface equipped with some pairwise disjoint arcs embedded in its boundary. In this paper, we extend the notion of character varieties to marked surfaces, in such a way that they have a nice behaviour for the operation of gluing two boundary arcs together. These stated character varieties are affine Poisson varieties which coincide with the Culler-Shalen character varieties when the surface is unmarked and are closely related to the Fock-Rosly and Alekseev-Kosmann-Malkin-Meinrenken constructions in the marked case. These Poisson varieties are the classical moduli spaces underlying stated skein algebras and share similar properties. In particular, stated character varieties admit triangular decompositions, associated to triangulations of the surface. We identify the Zariski tangent spaces of these varieties with some twisted groupoid cohomological groups and provide a generalization of Goldman's formula for the Poisson bracket of curve functions in terms of intersection form in homology.

Paper Structure

This paper contains 27 sections, 35 theorems, 148 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Stated character varieties
  3. Turning $\operatorname{Map}(X,G)$ into an affine scheme
  4. Definition of stated character varieties
  5. Stabilizer
  6. Curve functions
  7. Discrete models
  8. The gluing formula
  9. Triangular decompositions
  10. Twisted groupoid (co)homologies and the tangent spaces
  11. Twisted groupoid (co)homologies
  12. Comparison with standard twisted group (co)homology
  13. Relation with the (co)tangent spaces of the stated character varieties
  14. Discrete versions of twisted cohomology
  15. The intersection form
  16. ...and 12 more sections

Key Result

Theorem 1.3

Figures (6)

  • Figure 1: A finite presentation for the fundamental groupoid of the triangle.
  • Figure 4: Two elementary moves for pairs of geometric representatives of a pair of curves.
  • Figure 5: The marked surface $\mathbf{\Sigma}_{a\circledast b}$ is obtained from $\mathbf{\Sigma}$ by gluing a triangle $\mathbb{T}$. The figure illustrates how the embedding $\iota: \Sigma \hookrightarrow \Sigma_{a \circledast b}$ acts on curves.
  • Figure 6: On the left, the graph $\Gamma(4,1)$. On the right, a graph $\Gamma$ and a contracted graph $\Gamma(v)$.
  • Figure 7: Ten different configurations for a pair of oriented edges in some ciliated graph.
  • ...and 1 more figures

Theorems & Definitions (108)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4: Triangular decomposition of character varieties
  • Theorem 1.5
  • Theorem 1.6: Generalized Goldman formula
  • Remark 1
  • Definition 2.1
  • Remark 2
  • Lemma 2.2
  • ...and 98 more