Formality, Alexander invariants, and a question of Serre

Dimca, Alexandru; Papadima, Stefan; Suciu, Alexander I.

Formality, Alexander invariants, and a question of Serre

Authors

Dimca, Alexandru

Papadima, Stefan

Suciu, Alexander I.

Description

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.
43 pages, updated references