Resonance webs of hyperplane arrangements (Q2906435)

Let \({\mathcal A} \) be a hyperplane arrangement in \(P^n\), and let the complement of \({\mathcal A}\) be denoted by \(M\). Denote the torsion free cohomology ring of \(M\) by \(H^*(M, Z)\). Given \(a\in H^1(M, C)\), consider the complex given by multiplication by \(a\), where \(h^i: H^i(M) \to H^{i+1}(M)\). The resonance varieties of \(M\) are given by \(R^i(M) = \{a \in H^1(M) \mid h^i(H^*(M), a) \neq 0\}\). The first resonance variety was studied by \textit{M. Falk} and \textit{S. Yuzvinsky} [Compos. Math. 143, No. 4, 1069--1088 (2007; Zbl 1122.52009)]. For each irreducible component \(\Sigma\) of the resonance variety \(R^1(M)\), there is a unique foliation \({\mathcal F}_\Sigma\) on \(P^n\) defined by the corresponding pencil of hypersurfaces. The author defines the resonance web \({\mathcal W}(A)\) which is the superposition of all the foliations \({\mathcal F}_\Sigma\) associated to irreducible components of \(R^1\) and studies its space of abelian relations. The main result determines the rank of the resonance webs for the braid arrangements.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14003].