Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $${{\mathscr {O}}}_X$$ O X -module $${{\mathscr {A}}}$$ 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 $${{\mathscr {A}}}$$ A by choosing a global section V of $${{\mathscr {A}}}$$ 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 $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E of a holomolorphic vector bundle $${{\mathscr {E}}}$$ E on a complex manifold. If V is a differential operator on $${{\mathscr {E}}}$$ E with scalar symbol, i.e, a global section of $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E , we associate with the pair $$({{\mathscr {E}}},V)$$ ( 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 ($${{\mathscr {E}}}=0$$ E = 0 ) case.

Nonabelian holomorphic Lie algebroid extensions

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild–Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.