Naturality properties and comparison results for topological and infinitesimal embedded jump loci

TL;DR

This paper develops a method using augmented commutative differential graded algebra models to analyze the local structure of representation varieties and cohomology jump loci, providing explicit descriptions in special geometric cases.

Contribution

It introduces a new approach to describe germs of representation varieties and jump loci via infinitesimal models, under certain geometric and algebraic conditions.

Findings

Explicit descriptions for germs of representation varieties around trivial representation.

Applicable to compact Kähler manifolds and hyperplane arrangement complements.

Provides a framework for comparing topological and infinitesimal embedded jump loci.

Abstract

We use augmented commutative differential graded algebra (ACDGA) models to study $G$-representation varieties of fundamental groups $\pi=\pi_1(M)$ and their embedded cohomology jump loci, around the trivial representation 1. When the space $M$ admits a finite family of maps, uniformly modeled by ACDGA morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of ${\rm Hom} (\pi,G)$ and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when $M$ is either a compact K\"ahler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group $G$ is a non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$.