Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

TL;DR

This paper explores the differential graded cohomology of holomorphic vector fields on complex manifolds, establishing isomorphisms with hypercohomology, and applies these results to conformal field theory and complex structure deformations.

Contribution

It generalizes the relationship between differential graded cohomology and hypercohomology for holomorphic vector fields, extending previous results and applying them to mathematical physics and deformation theory.

Findings

Calculated cohomology up to the singular cohomology of mapping spaces

Established an isomorphism between differential graded cohomology and hypercohomology

Extended results of Kawazumi on complex Gelfand-Fuks cohomology

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.