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Decorated Local Systems and Character Varieties

Benedetta Facciotti, Marta Mazzocco, Nikita Nikolaev

Abstract

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $Σ$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $Σ$, as the moduli space of representations of the fundamental groupoid $Π_1 (Σ)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $Σ$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.

Decorated Local Systems and Character Varieties

Abstract

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces with boundaries with values in . In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on , as the moduli space of representations of the fundamental groupoid , as the space of monodromy data and as character variety. By adding marked points to the boundary of in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.
Paper Structure (38 sections, 42 theorems, 58 equations, 2 figures)

This paper contains 38 sections, 42 theorems, 58 equations, 2 figures.

Table of Contents

  1. Review: Local Systems and Character Varieties
  2. Groupoids and Moduli Spaces
  3. Groupoids
  4. Moduli spaces
  5. Three examples of Lie groupoids
  6. Linear Local Systems
  7. Groupoid Representations and Holonomy
  8. Holonomy representation
  9. Character Variety
  10. Monodromy Data
  11. Surfaces with Marked Boundary
  12. Discrete Fundamental Groupoid
  13. Finite Presentation
  14. Decorated Local Systems
  15. Filtered Local Systems
  16. ...and 23 more sections

Key Result

Theorem 1

For any surface with marked boundary $(\Sigma,P)$, there are canonical equivalences of categories between the groupoids of decorated local systems, decorated representations of the fundamental groupoid $\Pi_1 (\Sigma)$, decorated representations of the discrete fundamental groupoid $\pi_1 (\Sigma, P for any $\square \in \left\{ \rm{Fi}, \rm{Fr},\rm{PFr} \right\}$ where $K^\rm{Fi}_P \mathrel{\matho

Figures (2)

  • Figure 1: In the top picture the main point is a primary point. In the bottom one, the main point is a secondary point.
  • Figure 2:

Theorems & Definitions (100)

  • Theorem
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Remark 1.6
  • Proposition 1.7
  • ...and 90 more