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

Abstract

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of [Formula: see text], branched over the singular Kummer surface, with the branch locus in [Formula: see text] 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 [Formula: see text] branched over [Formula: see text] points, combined with the theorem of Narasimhan–Ramanan which identifies [Formula: see text] with [Formula: see text]. The singular points of [Formula: see text] correspond to abelian representations, and we prove that each has a neighborhood in [Formula: see text] homeomorphic to a cone on [Formula: see text].