# Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds

## TL;DR

This paper investigates the local structure of representation varieties and cohomology jump loci of quasi-projective manifolds, establishing explicit descriptions for certain groups and depths, and highlighting limitations in more general cases.

## Contribution

It provides a detailed relationship between the germs of representation varieties and their infinitesimal models for specific groups and depths, with explicit decompositions.

## Key findings

- Explicit descriptions for $ extrm{SL}_2( ext{C})$ and Borel subgroup cases at depth 1.
- Identification of strict inclusions in more complex group or depth scenarios.
- Connection between algebraic and infinitesimal models of jump loci.

## Abstract

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan-Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\textrm{SL}_2(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\textrm{SL}_n(\mathbb{C})$ for some $n\ge 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.