Fundamental groups, Alexander invariants, and cohomology jumping loci

TL;DR

This paper surveys the relationships between cohomology jumping loci, Alexander invariants, and fundamental groups, highlighting new examples and applications in topology and algebraic geometry.

Contribution

It introduces new examples and applications of cohomology jumping loci and Alexander invariants, raising questions and conjectures about their geometric and algebraic properties.

Findings

The geometry of jump loci relates to formality and projectivity.

Applications to hyperplane arrangements and 3-manifolds.

New insights into the structure of right-angled Artin groups.

Abstract

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The geometry of these varieties is intimately related to the formality, (quasi-) projectivity, and homological finiteness properties of \pi_1(X). We illustrate this approach with various applications to the study of hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin groups.