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The space of generalized projective structures on a Riemann surface $\Sigma$of genus g with n marked points is the affine space over the cotangent bundleto the space of SL(N)-opers. It is a phase space of $W_N$-gravity on$\Sigma\times\mathbb{R}$.  This space is a generalization of the space of projective structures on theRiemann surface. We define the moduli space of $W_N$-gravity as a symplecticquotient with respect to the canonical action of a special class of Liealgebroids. This moduli space describes in particular the moduli space ofdeformations of complex structures on the Riemann surface by differentialoperators of finite order, or equivalently, by a quotient space of Volterraoperators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids,because they are constructed by means of AGD bivector on the space of opersrestricted on a circle. The AGD-algebroids are particular case of Liealgebroids related to a Poisson sigma-model. The moduli space of thegeneralized projective structure can be described by cohomology of aBRST-complex.

Lie Algebroids and generalized projective structures on Riemann surfaces