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

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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 independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.