  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.
