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

Măcinic, Daniela; Papadima, Ştefan; Popescu, Clement; Suciu, Alexander

doi:10.1090/tran/6799

Flat connections and resonance varieties: From rank one to higher ranks
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by Daniela Anca Măcinic, Ştefan Papadima, Clement Radu Popescu and Alexander I. Suciu PDF

Trans. Amer. Math. Soc. 369 (2017), 1309-1343 Request permission

Abstract:

Given a finitely generated group $\pi$ and a linear algebraic group $G$, the representation variety $\mathrm {Hom}(\pi ,G)$ 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 $\pi$ is either an Artin group or the fundamental group of a smooth, quasi-projective variety.

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