A GEOMETRICAL APPROACH TO SUPER W-INDUCED GRAVITIES IN 2 DIMENSIONS

A geometrical study of supergravity defined on (1/1) complex superspac e is presented. This approach is based on the introduction of generali zed superprojective structures extending the notions of super Riemann geometry to a kind of super W-Riemann surfaces. On these surfaces a co nnection is constructed. The zero curvature condition leads to the sup er Ward identities of the underlying supergravity. This is accomplishe d through the symplectic form linked to the (super)symplectic manifold of all super gauge connections. The BRST algebra is also derived from the knowledge of the super W-symmetries which are the gauge transform ations of the vector bundle canonically associated to the generalized superprojective structures. We obtain the possible consistent BRST (su per)anomalies and their cocycles related by the descent equations. Fin ally we apply our considerations to The case of supergravity.