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

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

2001-01-08

arxiv.org/abs/math/0101057v1

Mathematics

Complex Variables

8 pages, no figures, Latex2e

Scientific paper

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.

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