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

Abstract

Let $\varSigma$ be an open Riemann surface and Hol($\varSigma$) be the Lie algebra of holomorphic vector fields on $\varSigma$. We fix a projective structure (i.e. a local $\text{SL}_2(\mathbf{C})$-structure) on $\varSigma$. We calculate the first group of cohomology of Hol($\varSigma$) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $\text{sl}_2(\mathbf{C})$. The result is independent on the choice of the projective structure. We give explicit formulas of 1-cocycles generating this cohomology group.