Polyanalytic forms on compact Riemann surfaces

A sheaf of differentials on a compact Riemann surface supplied with a projective structure is said to be $n$-analytic if in a local projective coordinate the sections of the sheaf satisfy the differential equation $\partial^nf/\partial\overline z^n=0$. For the projective structure induced by a covering mapping from the disk, an explicit characterization of the space of cross-sections and of the space of first cohomologies of the $n$-analytic sheaf is given in terms of known spaces of sections of certain holomorphic sheaves.