Abelian duality and propagation of resonance

TL;DR

This paper investigates how abelian duality properties influence the propagation of cohomology jump loci, establishing new links between algebraic duality, homological algebra, and geometric group theory.

Contribution

It introduces the concept of abelian duality spaces and the EPY property, connecting them to the propagation of characteristic and resonance varieties.

Findings

Propagation of jump loci is established for various classes of spaces.

Cohen-Macaulay condition is shown to be significant in this context.

Applications include arrangements, hyperplane complements, and Artin groups.

Abstract

We explore the relationship between a certain "abelian duality" property of spaces and the propagation properties of their cohomology jump loci. To that end, we develop the analogy between abelian duality spaces and those spaces which possess what we call the "EPY property." The same underlying homological algebra allows us to deduce the propagation of jump loci: in the former case, characteristic varieties propagate, and in the latter, the resonance varieties. We apply the general theory to arrangements of linear and elliptic hyperplanes, as well as toric complexes, right-angled Artin groups, and Bestvina-Brady groups. Our approach brings to the fore the relevance of the Cohen-Macaulay condition in this combinatorial context.