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

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 ${\text{SL}}_2(\mathbb{C})$ 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 $G={\text{SL}}_n(\mathbb{C})$ for some $n\ge 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

36 pages; accepted for publication in Journal of the Institute of Mathematics of Jussieu

References in corpus (5)

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Flat connections and resonance varieties: from rank one to higher ranks
• The topology of compact Lie group actions through the lens of finite models
• On the geometry and topology of partial configuration spaces of Riemann surfaces
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Cited by in corpus (2)

• The topology of compact Lie group actions through the lens of finite models
• Naturality properties and comparison results for topological and infinitesimal embedded jump loci