  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 bundle
to 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 the
Riemann surface. We define the moduli space of $W_N$-gravity as a symplectic
quotient with respect to the canonical action of a special class of Lie
algebroids. This moduli space describes in particular the moduli space of
deformations of complex structures on the Riemann surface by differential
operators of finite order, or equivalently, by a quotient space of Volterra
operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids,
because they are constructed by means of AGD bivector on the space of opers
restricted on a circle. The AGD-algebroids are particular case of Lie
algebroids related to a Poisson sigma-model. The moduli space of the
generalized projective structure can be described by cohomology of a
BRST-complex.
