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The module of logarithmic \(p\)-forms of a locally free arrangement. (English) Zbl 1047.14007

Summary: For an essential, central hyperplane arrangement \({\mathcal A}\subseteq V\simeq k^{n+1}\) we show that \(\Omega^1({\mathcal A})\) (the module of logarithmic one forms with poles along \({\mathcal A})\) gives rise to a lcoally free sheaf on \(\mathbb{P}^n\) if and only if, for all \(X\in L_{\mathcal A}\), the intersection lattice of \({\mathcal A}\), with \(\text{rank}\,X< \dim V\), the module \(\Omega^1 ({\mathcal A}_X)\) is free. Motivated by a result of L. Solomon and H. Terao [Adv. Math. 64, 305–325 (1987; Zbl 0625.05001)], we give a formula for the Chern polynomial of a bundle \({\mathcal E}\) on \(\mathbb{P}^n\) in terms of the Hilbert series of \(\bigoplus_{m\in\mathbb{Z}} H^0(\mathbb{P}^n,\bigwedge{\mathcal E}(m))\). As a corollary, we prove that if the sheaf associated to \(\Omega^1({\mathcal A})\) is locally free, then the Poincaré polynomial of \({\mathcal A}\), \(\pi({\mathcal A},t)\), is essentially the Chern polynomial. If \(\Omega^1({\mathcal A})\) has projective dimension one and is locally free, we give a minimal free resolution for \(\Omega^p\) and how that \(\bigwedge^p\Omega^1({\mathcal A})\simeq \Omega^p({\mathcal A})\), generalizing results of L. L. Rose and H. Terao [J. Algebra 136, 376–400 (1991; Zbl 0732.13010)] on generic arrangements.

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14C20 Divisors, linear systems, invertible sheaves
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
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