  If \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,\k),
viewed as a module over the exterior algebra E on \A: \theta_k(G) = \dim_\k
Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. 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 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 conjectured lower bound
for the Chen ranks of an arbitrary arrangement \A. Finally, we show that there
is a polynomial P(t) of degree equal to the dimension of R^1(\A), such that
\theta_k(G) = P(k), for k sufficiently large.
