Naturality properties and comparison results for 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 $\mathrm{Hom}(\pi ,G)$ 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 Kähler 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 ${\mathfrak{sl}}_2(\mathbb{C})$.

44 pages; accepted for publication in Advances in Mathematics

References in corpus (4)

• Deformation theory via differential graded Lie algebras
• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Flat connections and resonance varieties: from rank one to higher ranks
• Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds

Cited by in corpus (2)

• The topology of compact Lie group actions through the lens of finite models
• Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds