Canonical extensions of local systems

TL;DR

This paper introduces a method to extend local systems on complex manifolds as analytic spaces, particularly when the boundary has normal crossing singularities and the monodromy is unipotent, using Deligne's canonical extension.

Contribution

It proposes a novel extension of local systems as analytic spaces via the closure inside Deligne's canonical extension, with normalization being locally toric.

Findings

Extension exists as an analytic space with boundary behavior controlled by monodromy.

Normalization of the extended space is locally toric.

Applicable when the boundary divisor has normal crossings and monodromy is unipotent.

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.