Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

TL;DR

This paper explores the connection between geometric invariants of spaces and the homology jump loci, providing computable bounds and applications to various geometric and topological contexts.

Contribution

It introduces new bounds for Bieri-Neumann-Strebel-Renz invariants using exponential tangent cones and resonance varieties, linking geometric and homological properties.

Findings

Derived upper bounds for invariants using tangent cones

Connected invariants to resonance varieties in cohomology

Applied results to toric complexes, Artin kernels, and Kähler manifolds

Abstract

We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as K\"ahler and quasi-K\"ahler manifolds.