Jump loci in the equivariant spectral sequence

TL;DR

This paper investigates the relationship between homology jump loci and resonance varieties in the context of equivariant spectral sequences, revealing conditions under which certain finiteness properties hold for Alexander invariants.

Contribution

It establishes a connection between jump loci and resonance varieties in equivariant spectral sequences and shows vanishing resonance implies finiteness of Alexander invariants.

Findings

Vanishing resonance implies finiteness of Alexander invariants.

Vanishing resonance is a Zariski open condition.

Relates jump loci to resonance varieties in spectral sequences.

Abstract

We study the homology jump loci of a chain complex over an affine \k-algebra. When the chain complex is the first page of the equivariant spectral sequence associated to a regular abelian cover of a finite-type CW-complex, we relate those jump loci to the resonance varieties associated to the cohomology ring of the space. As an application, we show that vanishing resonance implies a certain finiteness property for the completed Alexander invariants of the space. We also show that vanishing resonance is a Zariski open condition, on a natural parameter space for connected, finite-dimensional commutative graded algebras.