Leray Spectral Sequence for Complements of Certain Arrangements of Smooth Submanifolds

Abstract

Let X be a complex algebraic manifold. Let U be the complement of a configuration of submanifolds. We study the Leray spectral sequence of the inclusion U ↪ X computing the cohomology of U. Under some condition posed on the intersections of submanifolds we show that the Leray spectral sequence degenerates on E3. This result generalizes well known properties of hyperplane arrangements. The main cause which rigidifies the spectral sequence is the weight filtration in cohomology.