### Abstract

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

Original language | English (US) |
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Pages (from-to) | 33-40 |

Number of pages | 8 |

Journal | Tokyo Journal of Mathematics |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface.** / Bouarroudj, Sofiane; Hichem, Gargoubi.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 25, no. 1, pp. 33-40. https://doi.org/10.3836/tjm/1244208934

}

TY - JOUR

T1 - Projectively Invariant Cocycles of Holomorphic Vector Fields on an Open Riemann Surface

AU - Bouarroudj, Sofiane

AU - Hichem, Gargoubi

PY - 2002/1/1

Y1 - 2002/1/1

N2 - Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL2(C)-structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl2(C). The result is independent on the choice of the projective structure. We give explicit formulas of 1-cocycles generating this cohomology group.

AB - Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL2(C)-structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl2(C). The result is independent on the choice of the projective structure. We give explicit formulas of 1-cocycles generating this cohomology group.

UR - http://www.scopus.com/inward/record.url?scp=0011567824&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011567824&partnerID=8YFLogxK

U2 - 10.3836/tjm/1244208934

DO - 10.3836/tjm/1244208934

M3 - Article

AN - SCOPUS:0011567824

VL - 25

SP - 33

EP - 40

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -