Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

Abstract

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.