2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140221We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between a set of decorations of an ideal triangulation of $S$ and a subset of the PU(2,1)-representation variety of $π_1(S)$.23 pages, 2 figuresDifferential Geometry57S30; 51M10; 57M50text