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