On the traceless SU(2) character variety of the 6-punctured 2-sphere

TL;DR

This paper describes the structure of the traceless SU(2) character variety of a 6-punctured sphere as a branched cover of complex projective space, revealing its singularities and their local topology.

Contribution

It establishes a geometric description of the character variety as a branched cover over CP^3 and analyzes the local topology near singular points.

Findings

Character variety is a 2-fold branched cover of CP^3.

Singular points correspond to abelian representations.

Neighborhoods of singular points are cones over S^2×S^3.

Abstract

We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ character varieties by the 2-fold branched cover $F_{n-1}\to S^2$ branched over $2n$ points, combined with the theorem of Narasimhan-Ramanan which identifies $R(F_2)$ with ${\mathbb{C}} P^3$. The singular points of $R(S^2,6)$ correspond to abelian representations, and we prove that each has a neighborhood in $R(S^2,6)$ homeomorphic to a cone on $S^2\times S^3$.