Nonabelian cohomology jump loci from an analytic viewpoint

TL;DR

This paper explores the structure of cohomology jump loci in topological spaces and their algebraic models, revealing their geometric properties and relationships through an analytic and algebraic perspective.

Contribution

It introduces a new analytic approach to understanding nonabelian cohomology jump loci using CDG algebras and minimal models, establishing their local structure and connections to algebraic geometry.

Findings

Representation varieties' germs are determined by Sullivan's minimal models.

Up to a certain degree, topological and algebraic jump loci share the same local structure.

When CDG algebras have positive weights, jump loci components are affine subtori or rational linear subspaces.

Abstract

For a topological space, we investigate its cohomology support loci, sitting inside varieties of (nonabelian) representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its cohomology jump loci, sitting inside varieties of (algebraic) flat connections. We prove that the analytic germs at the origin 1 of representation varieties are determined by the Sullivan 1-minimal model of the space. Under mild finiteness assumptions, we show that, up to a degree $q$, the two types of jump loci have the same analytic germs at the origins, when the space and the algebra have the same $q$-minimal model. We apply this general approach to formal spaces (for which we establish the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic jump loci: up to degree $q$, all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic cohomology jump loci, up to degree $q$, into their topological counterpart.