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Let $\Sigma$ be an open Riemann surface and $Hol (\Sigma)$ be the Lie algebraof holomorphic vector fields on $\Sigma.$ We fix a projective structure (i.e. alocal $SL_2(C)-$structure) on $\Sigma.$ We calculate the first group ofcohomology of $Hol(\Sigma)$ with coefficients in the space of linearholomorphic operators acting on tensor densities, vanishing on the Lie algebra$SL_2 (C).$ The result is independant on the choice of the projectivestructure. We give explicit formulae of 1-cocycles generating this cohomologygroup.

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