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

Given a finitely-generated group $π$ and a linear algebraic group $G$, the representation variety Hom$(π,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $G$. Its algebraic counterpart, 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 $\mathfrak{g}$. We establish here a number of 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 $π$ is either an Artin group, or the fundamental group of a smooth, quasi-projective variety.

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