Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free OX-module A which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with A by choosing a global section V of A, and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid DE of a holomolorphic vector bundle E on a complex manifold. If V is a differential operator on E with scalar symbol, i.e, a global section of DE, we associate with the pair (E, V) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted (E= 0) case.