  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.
