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

Schenck, Henry; Suciu, Alexander

doi:10.1090/S0002-9947-05-03853-5

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
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by Henry K. Schenck and Alexander I. Suciu PDF

Trans. Amer. Math. Soc. 358 (2006), 2269-2289 Request permission

Abstract:

If $\mathcal A$ 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 $A=H^*(X,\Bbbk )$, viewed as a module over the exterior algebra $E$ on $\mathcal A$: \[ \theta _k(G) = \dim _{\Bbbk }\operatorname {Tor}^E_{k-1}(A,\Bbbk )_k, \quad \text {for $k\ge 2$}, \] where $\Bbbk$ is a field of characteristic $0$. 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 $\mathcal R^{1}(\mathcal A)$. 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 $\mathcal R ^{1}(\mathcal A)$ and a localization argument, we establish the inequality \[ \theta _k(G) \ge (k-1) \sum _{r\ge 1} h_r \binom {r+k-1}{k}, \quad \text {for $k\gg 0$}, \] for arbitrary $\mathcal A$. Finally, we show that there is a polynomial $\mathrm {P}(t)$ of degree equal to the dimension of $\mathcal R^1(\mathcal A)$, such that $\theta _k(G) = \mathrm {P}(k)$, for all $k\gg 0$.

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