  We formulate and solve the analog of the universal Conformal Ward Identity
for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and
present a rigorous invariant formulation of the chiral sector in the induced
two-dimensional gravity on higher genus Riemann surfaces. Our construction of
the action functional uses various double complexes naturally associated with a
Riemann surface, with computations that are quite similar to descent
calculations in BRST cohomology theory. We also provide an interpretation for
the action functional in terms of the geometry of different fiber spaces over
the Teichm\"{u}ller space of compact Riemann surfaces of genus $g>1$.
