On compactifications of the SL(2,C) character varieties of punctured surfaces

TL;DR

This work advances the understanding of compactifications of $\mathrm{SL}(2,\mathbb C)$ character varieties for punctured surfaces by developing two complementary constructions: triangulation-based absolute/relative compactifications with toric boundary divisors and a uniform relative GIT-based compactification framework via $\overline{\mathrm{SL}}(2,\mathbb C)$. It proves that the boundary divisors are toric and that the dual boundary complex of any positive-dimensional relative character variety is a sphere, establishing the geometric $P=W$-type sphere property. The approach yields explicit moment-polytope descriptions, monodromy data, and a uniform parametrization of conjugacy classes, enabling a coherent analysis across the base space $\mathbb C^n$ and across mutations of triangulations. By also incorporating anti-holomorphic involutions, the paper describes real loci corresponding to $\mathrm{SL}(2,\mathbb R)$ and $\mathrm{SU}(2)$, providing compactifications of elliptic and SU(2) loci and demonstrating their compatibility with the complex picture. The results have implications for NAHC, the skein-algebra perspective, and the broader study of compactifications in character-variety theory and cluster-algebra frameworks.

Abstract

This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the $\SL(2,\C)$-character variety of an $n$-punctured Riemann surface without boundary. We study a class of projective compactifications determined by ideal triangulations of the surface and prove explicit results concerning the boundary divisors of these compactifications. Notably, we establish that the boundary divisors are toric varieties and confirm a well-known conjecture asserting that the (dual) boundary complex of any (positive dimensional) relative character variety is a sphere. In a different vein, we enhance and streamline Komyo's compactification method, which utilizes a projective compactification of $\SL(2,\C)$ to compactify the (relative) character varieties. Specifically, we construct a uniform relative compactification over the base space of $\C^n$ and determine its monodromy, addressing a question posed by Simpson.

Figures (18)

• Figure 1: A triangle folded on the arc $c$
• Figure 2: Line segments on the edges of a triangle $\{e,e',e"\}$ for $v\in\Lambda$ satisfying $(v_e,v_{e'},v_{e"})=(4,2,2)$.
• Figure 3: Left- two line segments crossing in a triangle. Middle- the resolution that creates a bigon. Right- the resolution that defines the geometric sum.
• Figure 4: Red and blue curves generate $\Lambda\subset \mathbb N^3$. Black lines in the rectangle are the edges of the triangulation.
• Figure 5: In each diagram red and blue curves generate $\Lambda\subset \mathbb N^6$. Black lines are the edges of a triangulation, with the half-open edges meeting at the fourth punctured point located at $\infty$.

Theorems & Definitions (68)

• Conjecture 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.10
• Remark 1.11