Canonical extensions of local systems

Abstract

A local system H on a complex manifold M can be viewed in two ways--either as a locally free sheaf, or as a union of covering spaces T = T(H). When M is an open set in a bigger manifold, the local system will generally not extend, because of local monodromy. This paper proposes an extension of the local system as an analytic space, in the case when the complement of M has normal crossing singularities, and the local system is unipotent along the boundary divisor. The analytic space is obtained by taking the closure of T inside the total space of Deligne's canonical extension of the associated vector bundle. It is not normal, but its normalization is locally toric.