Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface

Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL²(C)-structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl²(C). The result is independent on the choice of the projective structure. We give explicit formulas of 1-cocycles generating this cohomology group.