Triangular decomposition of character varieties

A marked surface is a compact oriented surface equipped with some pairwise disjoint arcs embedded in its boundary. In this paper, we extend the notion of character varieties to marked surfaces, in such a way that they have a nice behaviour for the operation of gluing two boundary arcs together. These stated character varieties are affine Poisson varieties which coincide with the Culler–Shalen character varieties when the surface is unmarked and are closely related to the Fock–Rosly and Alekseev–Kosmann–Malkin–Meinrenken constructions in the marked case. These Poisson varieties are the classical moduli spaces underlying stated skein algebras and share similar properties. In particular, stated character varieties admit triangular decompositions, associated to triangulations of the surface. We identify the Zariski tangent spaces of these varieties with some twisted groupoid cohomological groups and provide a generalization of Goldman’s formula for the Poisson bracket of curve functions in terms of intersection form in homology.