An (analytic) approach to the uniformization of a compact Riemann surface of genus greater than one is to look at metrics of constant negative curvature. Such metrics (more precisely, the condition that characterizes them) admit a generating functional, extending the classical Liouville's one. This functional is naturally interpreted as the square of the metrized holomorphic tangent bundle in a suitably defined hermitian-holomorphic Deligne cohomology group. For a pair of line bundles on the Riemann surface, it generalizes Deligne's tame symbol. If time permits, we will also outline relations with group cohomology for Kleinian groups and volume calculations in hyperbolic 3-space.