The local cohomology of vector fields

TL;DR

This paper computes the local cohomology of vector fields on manifolds, connecting smooth and holomorphic cases, and constructs explicit cocycle representatives in Gelfand--Fuks cohomology.

Contribution

It provides a unified computation of local cohomology for vector fields in both smooth and holomorphic contexts, including explicit cocycle representatives.

Findings

Recovers diagonal cohomology in the smooth case.

Relates to recent work on Lie algebra cohomology of vector fields.

Constructs explicit cocycle representatives via descent.

Abstract

We compute the local cohomology of vector fields on a manifold. In the smooth case this recovers the diagonal cohomology studied in work of Losik, Guillemin, Fuks and others. In the holomorphic case this cohomology has recently appeared in work of Hennion and Kapranov in their study of the Lie algebra cohomology of vector fields on a complex manifold. Additionally, we construct explicit representatives for cocycles in Gelfand--Fuks cohomology via descent.