# 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.

## Key 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.