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

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

2005-02-21

arxiv.org/abs/math/0502438v2

Trans. Amer. Math. Soc. 358 (2006), no. 5, 2269-2289

Mathematics

Commutative Algebra

21 pages; final version

Scientific paper

10.1090/S0002-9947-05-03853-5

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.

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