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

TL;DR

This paper provides explicit formulas for cocycles of holomorphic vector fields with values in lambda-densities, utilizing affine and projective connections, and extends known circle cocycles to open Riemann surfaces.

Contribution

It introduces explicit formulas for the generators of second cohomology groups of holomorphic vector fields with lambda-densities, generalizing circle cocycles to open Riemann surfaces.

Findings

Explicit formulas for cocycles in terms of affine and projective connections

Globalization of circle cocycles to open Riemann surfaces

Enhanced understanding of cohomology of holomorphic vector fields

Abstract

We give explicit formulae for the generators of H^2(Hol(\Sigma_r,{\cal F}_{\lambda}(\Sigma_r)) in terms of affine and projective connections. This is done using the cocycles of V. Ovsienko and C. Roger for the case of the circle and globalizing them to an open Riemann surface \Sigma_r.