If A. is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 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,k), viewed as a module over the exterior algebra E on A: θ k(G) = dim k Tor k-1 E (A,k) k, for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ k(G) = (k - 1) ∑ r≥1 h r ( r+k-1 k, where h r is the number of r-dimensional components of the projective resonance variety R 1(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 R 1(A) and a localization argument, we establish the inequality θ k(k-1) ∑ r>1 hr ( k r+k-1), f k ≫ 0 for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R 1l(A), such that θ k(G) = P(k), for all k ≫ 0.
ASJC Scopus subject areas
- Applied Mathematics