Resonance, Linear Syzygies, Chen Groups, and the Bernstein-gelfand-gelfand Correspondence

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H(X, k), viewed as a module over the exterior algebra E on A: θk(G) = dimk Tor E k−1(A, k)k , for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θk(G) = (k− 1) ∑ r≥1 hr ( r+k−1 k ) , where hr is the number of r-dimensional components of the projective resonance variety R1(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 R1(A) and a localization argument, we establish the inequality θk(G) ≥ (k − 1) ∑ r≥1 hr (r + k − 1 k ) , for k ≫ 0, for arbitrary A. Finally, we show that there is a polynomial P (t) of degree equal to the dimension of R1(A), such that θk(G) = P (k), for all k ≫ 0.