  An effective action is obtained for the $N=1$, $2D-$induced supergravity on a
compact super Riemann surface (without boundary) $\hat\Sigma$ of genus $g>1$,
as the general solution of the corresponding superconformal Ward identity. This
is accomplished by defining a new super integration theory on $\hat\Sigma$
which includes a new formulation of the super Stokes theorem and residue
calculus in the superfield formalism. Another crucial ingredient is the notion
of polydromic fields. The resulting action is shown to be well-defined and free
of singularities on $\sig$. As a by-product, we point out a morphism between
the diffeomorphism symmetry and holomorphic properties.
