[Paper Review] On compactifications of character varieties of $n$-punctured projective line

Arata Komyo|arXiv (Cornell University)|Jul 30, 2013

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper constructs compactifications of $SL_2(\mathbb{C})$-character varieties for the $n$-punctured projective line and verifies Simpson's conjecture on the boundary divisor's configuration by analyzing the boundary complex via blow-ups and simplicial decompositions. It proves the boundary complex is a simplicial decomposition of a sphere, confirming the homotopy type conjectured by Simpson.

ABSTRACT

In this paper, we construct compactifications of $SL_2(\mathbb{C})$-character varieties of $n$-punctured projective line and study the boundary divisor of the compactifications. This study is motivated by the conjecture for the configuration of the boundary divisor, due to C. Simpson. We verify the conjecture for a few examples.

Motivation & Objective

• To construct smooth compactifications of $SL_2(\mathbb{C})$-character varieties for the $n$-punctured projective line with normal crossing boundary divisors.
• To verify Simpson's conjecture on the homotopy type of the boundary complex of such compactifications.
• To analyze the configuration of irreducible components of the boundary divisor using blow-ups and simplicial decomposition techniques.
• To establish that the boundary complex is a simplicial decomposition of a sphere, confirming the conjectured homotopy type.

Proposed method

• Constructs a compactification via iterated blow-ups along strata of the boundary divisor to resolve singularities and ensure normal crossings.
• Uses the boundary complex $\Delta(D^{{\it B}}_{g,\boldsymbol{\mu}})$ to encode the intersection pattern of irreducible components of the boundary divisor.
• Applies non-abelian Hodge theory and the Hitchin fibration to relate the Dolbeault and Betti moduli spaces and their compactifications.
• Analyzes intersections of boundary components (e.g., $E_i \cap E_j$, $E_i \cap \text{ex}_{k,k+2}^{\pm}$) to determine irreducibility and non-emptiness.
• Employs a geometric gluing procedure to construct 2- and 3-dimensional simplices from irreducible intersections of components.
• Demonstrates that the resulting boundary complex is a simplicial decomposition of $S^3$ by embedding vertexes on $S^2 \subset \mathbb{R}^4$ and attaching $\text{ex}_1, \text{ex}_2$ at poles.

Experimental results

Research questions

• RQ1 Does the boundary complex of a compactified $SL_2(\mathbb{C})$-character variety for the $n$-punctured projective line have the homotopy type of a sphere?
• RQ2 Can the configuration of the boundary divisor be described via a simplicial decomposition that matches the conjectured structure?
• RQ3 Are the intersections of boundary components irreducible and non-empty in a way that supports a simplicial structure?
• RQ4 How do blow-ups along orbits of special points resolve singularities and facilitate the construction of the boundary complex?

Key findings

• The boundary complex $\Delta(D^{{\it B}}_{g,\boldsymbol{\mu}})$ of the compactification is a simplicial decomposition of $S^3$, confirming the conjectured homotopy type.
• The intersections $\text{ex}_{1,3}^{+} \cap E_2$, $\text{ex}_{1,3}^{-} \cap E_4$, $\text{ex}_{2,4}^{+} \cap E_1$, and $\text{ex}_{2,4}^{-} \cap E_3$ are nonempty and irreducible, supporting the simplicial structure.
• The intersections $E_1 \cap E_2 \cap \text{ex}_{1,3}^{+}$ and $E_2 \cap E_3 \cap \text{ex}_{1,3}^{+}$ are irreducible after blow-up along $E_{1,3}^{+}$, enabling triangle gluing.
• The 3-tuples $E_i \cap E_{i+1} \cap \text{ex}_j$ and $E_{k,k+2}^{\pm} \cap \text{ex}_j$ are nonempty and irreducible, allowing gluing of 3-simplices.
• The 4-tuples containing $\text{ex}_1$ or $\text{ex}_2$ are glued together, and the complex forms a triangulation of $S^3$.
• The boundary complex is independent of the compactification choice, confirming the homotopy invariance of the structure.