Koszul modules, holonomy Lie algebras, and resonance of groups and CDGAs

TL;DR

This paper introduces a Koszul-theoretic framework linking Alexander invariants, holonomy Lie algebras, and resonance varieties, providing explicit formulas and applications to group invariants and geometric structures.

Contribution

It develops a new Koszul linearization approach to compare classical and infinitesimal invariants, with explicit formulas and broad applications.

Findings

Koszul modules $\

First Koszul module $\

Abstract

We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The central mechanism is Koszul linearization, which replaces nonlinear equivariant constructions with functorial algebraic objects defined from a CDGA. To a connected CDGA $(A,d)$ with finite-dimensional $H^1(A)$ we associate Koszul modules $\mathcal{B}_i(A)$ over the symmetric algebra on $H_1(A)$. We prove that the first Koszul module $\mathcal{B}_1(A)$ is isomorphic to the infinitesimal Alexander invariant of the holonomy Lie algebra $\mathfrak{h}(A)$, yielding explicit formulas for holonomy Chen ranks. We establish a tangent cone theorem for resonance varieties, showing that cohomology controls their first-order behavior at the origin. For finitely generated groups admitting 1-finite 1-models, we prove that classical Alexander invariants agree with Koszul invariants after completion and passage to associated graded objects, and that Chen ranks are determined by the model. Applications include Chen rank computations for 2-step nilpotent Lie algebras and pure elliptic braid groups, partial formality results for Sasakian manifolds, and a general framework for detecting non-formality of spaces, groups, and maps.