Topology and geometry of cohomology jump loci

TL;DR

This paper explores the structure of cohomology jump loci, revealing how formality influences their geometry and providing new obstructions to certain groups being realized as fundamental groups of complex algebraic varieties.

Contribution

It introduces relative characteristic and resonance varieties, analyzes their local properties, and applies these findings to classical problems in algebraic topology and geometry.

Findings

Germs of V_k and R_k are analytically isomorphic for 1-formal groups

Tangent cone to V_k at 1 equals R_k, revealing rationality properties

Provides obstructions to realizing groups as fundamental groups of complex varieties

Abstract

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.