Suciu's Chen-rank formula conjecture from resonance

Let A{\mathcal{A}} be a hyperplane arrangement, let G=G(A)G=G({\mathcal{A}}) be its arrangement group, and let R1(A)R^{1}({\mathcal{A}}) be its first resonance variety. For each rr, let hrh_r be the number of components of R1(A)R^{1}({\mathcal{A}}) of dimension rr. The Chen ranks are defined by

θk(G):=ϕk(G/G),\theta_k(G):=\phi_k(G/G”),

where G=[G,G]G'=[G,G] and G=[G,G]G”=[G',G']. Suciu's conjecture. For k0k\gg 0,

θk(G)=(k1)r1hr(r+k1k).\theta_k(G)= (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}.

The conjecture relates the asymptotic Chen ranks of an arrangement group to the dimensions and multiplicities of the components of its first resonance variety. The source does not state a resolution.

Sources & referencesView supporting material

Primary source

Hal Schenck, “Hyperplane Arrangements: Computations and Conjectures”, arXiv:1101.0356 (2012).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.