Formality, Alexander invariants, and a question of Serre

TL;DR

This paper explores how formality influences Alexander invariants and characteristic varieties, providing new obstructions to 1-formality and insights into the realization of groups as fundamental groups of complex algebraic varieties.

Contribution

It demonstrates that the Alexander invariant's I-adic completion depends only on low-degree cup products, linking formality to geometric and algebraic obstructions.

Findings

Germs of characteristic and resonance varieties are analytically isomorphic at the origin.

New obstructions to 1-formality are established based on tangent cone analysis.

Applications include insights into arrangements, configuration spaces, and Artin groups.

Abstract

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We show that the I-adic completion of the Alexander invariant of a 1-formal group G is determined solely by the cup-product map in low degrees. It follows that the germs at the origin of the characteristic and resonance varieties of G are analytically isomorphic; in particular, the tangent cone to V_k(G) at 1 equals R_k(G). This provides new obstructions to 1-formality. 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.