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Metrics of constant negative curvature on a compact Riemann surface arecritical points of the Liouville action functional, which in recentconstructions is rigorously defined as a class in a Cech-de Rham complex withrespect to a suitable covering of the surface.  We show that this class is the square of the metrized holomorphic tangentbundle in hermitian-holomorphic Deligne cohomology. We achieve this byintroducing a different version of the hermitian-holomorphic Deligne complexwhich is nevertheless quasi-isomorphic to the one introduced by Brylinski inhis construction of Quillen line bundles. We reprove the relation with thedeterminant of cohomology construction.  Furthermore, if we specialize the covering to the one provided by a Kleinianuniformization (thereby allowing possibly disconnected surfaces) the same classcan be reinterpreted as the transgression of the regulator class expressed bythe Bloch-Wigner dilogarithm.Comment: AMSLatex + XYpic + geometry. Some substantial changes in exposition,  same results. Version to appear in Communications in Mathematical Physic

On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action