Koszul complexes and spectral sequences associated with Lie algebroids

TL;DR

This paper investigates spectral sequences related to Lie algebroids on complex manifolds or schemes, demonstrating their degeneration properties and introducing new twisted Koszul complexes for Atiyah algebroids.

Contribution

It introduces and analyzes spectral sequences for Lie algebroids, including a new twisted Koszul complex for Atiyah algebroids, and proves their degeneration under certain conditions.

Findings

Spectral sequence degenerates at the second page using Deligne's criterion.

The spectral sequence for the Atiyah algebroid degenerates at the first page in the untwisted case.

New twisted Koszul complex associated with differential operators on vector bundles.

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