Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If $\mathcal A$ is a complex hyperplane arrangement, with complement $X$, we show that the Chen ranks of $G=\pi_1(X)$ are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring $A=H^*(X,\Bbbk)$, viewed as a module over the exterior algebra $E$ on $\mathcal A$:

$\theta_k(G) = \dim_{\Bbbk}\operatorname{Tor}^E_{k-1}(A,\Bbbk)_k,$   for $k\ge 2$$,$

where $\Bbbk$ is a field of characteristic 0. The Chen ranks conjecture asserts that, for $k$ sufficiently large, $\theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$, where $h_r$ is the number of $r$-dimensional components of the projective resonance variety $\mathcal R^{1}(\mathcal A)$. Our earlier work on the resolution of $A$ over $E$ and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of $\mathcal R ^{1}(\mathcal A)$ and a localization argument, we establish the inequality

$\theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k},$   for $k\gg 0$$,$

for arbitrary $\mathcal A$. Finally, we show that there is a polynomial $\mathrm{P}(t)$ of degree equal to the dimension of $\mathcal R^1(\mathcal A)$, such that $\theta_k(G) = \mathrm{P}(k)$, for all $k\gg 0$.