Explicit Formulae for Cocycles of Holomorphic Vector Fields with values in λ Densities

The continuous cohomology of Lie algebras of C-vector fields has been studied by I. M. Gelfand, D. B. Fuks, R. Bott, A. Haefliger and G. Segal in some outstanding papers [4], [9], [1]. B. L. Feigin [2] and N. Kawazumi [11], whose work is continued in [18], studied Gelfand-Fuks cohomology of Lie algebras of holomorphic vector fields Hol(Σ) on an open Riemann surface. Kawazumi calculated the cohomology spaces H∗(Hol(Σ),Fλ(Σ)) of Hol(Σ) with values in the space of (holomorphic) λ-densities on Σ, using a well known theorem of Goncharova, cf [3]. He expressed the generators of the cohomology spaces in terms of the nowhere-vanishing holomorphic vector field ∂ which exists on open Riemann surfaces, trivializing the holomorphic tangent bundle. In this article, we give explicit formulae for the generators ofH(Hol(Σ),Fλ(Σ)) in terms of affine and projective connections. This is done using the cocycles which have been evidenced by V. Ovsienko and C. Roger in [13] and globalizing them by their transformation property. The main reason to look for explicit formulae is the search for a generalization of the Krichever-Novikov algebras [12] to semi-direct products of Hol(Σ) with Fλ(Σ), cf [13] for the case of V ect(S).