  We 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 $\pi_1(S)$.

Differential Geometry