Cohomology jump loci and absolute sets for singular varieties

TL;DR

This paper extends the concept of absolute subsets to singular varieties, demonstrating that rank one twisted cohomology jump loci are finite unions of translated subtori, with applications to Hodge theory and unitary local systems.

Contribution

It generalizes the notion of absolute sets to normal varieties and proves structural results about cohomology jump loci in this broader context.

Findings

Rank one twisted cohomology jump loci are finite unions of translated subtori.

The same holds for loci twisted by unitary local systems on projective varieties.

Interaction with Hodge structures reveals deeper geometric properties.

Abstract

We extend the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over a normal variety are a finite union of translated subtori. We show that the same holds for jump loci twisted by a unitary local system in the case where the underlying variety $X$ is projective with $H^1(X,\mathbb{Q})$ pure of weight one. Lastly, we study the interaction of these loci with Hodge theoretic data naturally associated to the representation variety of fundamental groups of smooth projective varieties.