Differential Graded Cohomology and Lie Algebras¶of Holomorphic Vector Fields

This article continues work of B. L. Feigin [5] and N. Kawazumi [15] on the Gelfand-Fuks cohomology of the Lie algebra of holomorphic vector fields on a complex manifold. As this is not always an interesting Lie algebra (for example, it is 0 for a compact Riemann surface of genus greater than 1), one looks for other objects having locally the same cohomology. The answer is a cosimplicial Lie algebra and a differential graded Lie algebra (well known in Kodaira–Spencer deformation theory). We calculate the corresponding cohomologies and the result is very similar to the result of A. Haefliger [12], R. Bott and G. Segal [2] in the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} vector fields. Applications are in conformal field theory (for Riemann surfaces), deformation theory and foliation theory.