Koszul complexes and spectral sequences associated with Lie algebroids

Abstract

We study some spectral sequences associated with a locally free ${\mathcal{O}}_X$-module $\mathcal{A}$ 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 $\mathcal{A}$ by choosing a global section $V$ of $\mathcal{A}$, 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 ${\mathcal{D}}_E$ 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 ${\mathcal{D}}_E$, 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

8 pages. To appear in São Paulo Journal of Mathematical Sciences

References in corpus (1)

• Lie algebroid cohomology as a derived functor