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

S. Bouarroudj; H. Gargoubi

paper

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

arXiv:math/0101057

Abstract

Let $Σ$ be an open Riemann surface and $H o l (Σ)$ be the Lie algebra of holomorphic vector fields on $Σ.$ We fix a projective structure (i.e. a local $S L_{2} (C) -$ structure) on $Σ.$ We calculate the first group of cohomology of $H o l (Σ)$ with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $S L_{2} (C) .$ The result is independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.

8 pages, no figures, Latex2e