Characteristic Varieties of Hypersurface Complements

TL;DR

This paper establishes divisibility relations for the characteristic varieties of hypersurface complements, linking global properties to local data and applying these results to Alexander modules and resonance varieties, especially in hyperplane arrangements.

Contribution

It introduces new divisibility results connecting global and local characteristic varieties and extends these to Alexander modules and resonance varieties in hyperplane arrangements.

Findings

Divisibility results for global characteristic varieties in terms of local data.

New finiteness results for Alexander modules of hypersurface complements.

Translation of divisibility results into resonance varieties for hyperplane arrangements.

Abstract

We give divisibility results for the (global) characteristic varieties of hypersurface complements expressed in terms of the local characteristic varieties at points along one of the irreducible components of the hypersurface. As an application, we recast old and obtain new finiteness and divisibility results for the classical (infinite cyclic) Alexander modules of complex hypersurface complements. Moreover, for the special case of hyperplane arrangements, we translate our divisibility results for characteristic varieties in terms of the corresponding resonance varieties.