Geometric and homological finiteness in free abelian covers

TL;DR

This paper explores the relationships between various invariants and cohomology support loci to understand finiteness properties of free abelian covers, with applications to complex varieties and hyperplane arrangements.

Contribution

It establishes connections between geometric, homological, and cohomological invariants, providing new insights into the finiteness properties of free abelian covers under certain conditions.

Findings

Resonance varieties determine finiteness properties in specific cases.

Translated components in characteristic varieties influence the invariants.

Applications to toric complexes and hyperplane arrangement complements.

Abstract

We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect the answer. We illustrate this theory in the setting of toric complexes, as well as smooth, complex projective and quasi-projective varieties, with special emphasis on configuration spaces of Riemann surfaces and complements of hyperplane arrangements.