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

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: H.K. Schenck, A.I. Suciu

Editorial: Transactions of the American Mathematical Society, 358 (5), p.2269-2289, 2006.

Rezumat:

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: heta_k(G) = dim_k Tor^E_{k-1}(A,k)_k, where k is a field of characteristic 0, and kge 2. The Chen ranks conjecture asserts that, for k sufficiently large, heta_k(G) =(k-1) sum_{rge 1} h_r inom{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 heta_k(G) = P(k), for k sufficiently large.

Cuvinte cheie: Chen groups, resonance varieties, BGG correspondence

URL: http://www.ams.org/tran/2006-358-05/S0002-9947-05-03853-5