Flat connections and resonance varieties: from rank one to higher ranks (Q2833021)

Let \(\pi\) be a finitely generated group and let \(G\) be a complex linear algebraic group. Then \(\mathrm{Hom}(\pi, G)\) is the resulting representation variety which has a natural filtration by the characteristic varieties associated to a rational representation of \(G\). In the rank \(1\) case, one finds the character group \(\mathbb T(\pi) = \mathrm{Hom}(\pi, \mathbb C^*)\) which is simply the Pontryagin dual of the abelianization of \(\pi\). In general, though, the \(G\)-representation variety is highly complicated, yet carries intricate information on the given group \(\pi\). The Kapovich-Millson universality theorem describes the complexity of representation varieties in a precise way: the analytic germs away from the trivial representation \(1\in\mathrm{Hom}(\pi,\mathrm{PSL}_2)\), where \(\pi\) runs through the family of Artin groups, are as complicated as arbitrary germs of affine varieties defined over \(\mathbb Q\).NEWLINENEWLINELet \(\mathfrak g\) be a Lie algebra. Its algebraic analogue of \(\mathrm{Hom}(\pi, G)\), the space of \(\mathfrak g\)-valued flat connections on a commutative, differential graded algebra \((A, d)\), admits a filtration by the resonance varieties associated to a representation of \(\mathfrak g\).NEWLINENEWLINEIn this paper the authors establish some results concerning the structure and qualitative properties of these embedded resonance varieties, with particular attention to the case when the rank \(1\) resonance variety decomposes as a finite union of linear subspaces. The general theory is illustrated in detail in the case when \(\pi\) is either an Artin group or the fundamental group of a smooth, quasi-projective variety.