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Hyperplane arrangements of Torelli type. (English) Zbl 1278.14027

Let \(D\) be a reduced divisor in the complex projective space \(\mathbb P^n,\) let \(\Omega_{\mathbb P^n}^p(\log D),\) \(p=0,\dots, n,\) be the sheaves of logarithmic differential \(p\)-forms, and \(\omega^p_D,\) \(p=0,\dots, n-1,\) the sheaves of regular meromorphic \(p\)-forms on the divisor. Then for all \(p\geq 0\) there exist exact sequences \(0\rightarrow \Omega_{\mathbb P^n}^p\longrightarrow\Omega_{\mathbb P^n}^p(\log D) \overset{\text{res}}\longrightarrow \omega^{p-1}_D \rightarrow 0\) (see [A. G. Aleksandrov, Adv. Sov. Math. 1, 211–246 (1990; Zbl 0731.32005)]) which are induced by the residue map of K. Saito [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)]. If \(D\) is a normal crossing divisor then \(\omega^0_D\) coincides with the sheaf \(n_*({\mathcal O}_{\widetilde{D}})\) of germs of weakly holomorphic functions on \(D,\) where \(n: \widetilde{D}\rightarrow D\) is the morphism of normalization. However, in general, \(n_*({\mathcal O}_{\widetilde{D}})\) is a proper subsheaf of \(\omega^0_D\) (see [D. Barlet, Lect. Notes Math. 670, 187–204 (1978; Zbl 0398.32009)]). Following to [I. V. Dolgachev, J. Math. Kyoto Univ. 47, No. 1, 35–64 (2007; Zbl 1156.14015)], let denote by \(\widetilde{\Omega}_{\mathbb P^n}(\log D) \subseteq \Omega_{\mathbb P^n}^1(\log D)\) the preimage of the sheaf \(n_*({\mathcal O}_{\widetilde{D}})\) under the residue map. By definition, an arrangement \(D\) of hyperplanes in \(\mathbb P^n\) is called semi-stable if the sheaf \(\widetilde{\Omega}_{\mathbb P^n}(\log D)\) is semi-stable. The set \(Z\) of the points corresponding to the hyperplanes of \(D\) in the dual projective space \({\check{\mathbb P} ^n}\) is called the dual configuration of the arrangement \(D = D_Z.\) Dolgachev conjectured that semi-stable arrangements with isomorphic sheaves \(\widetilde{\Omega}_{\mathbb P^n}(\log D_Z)\) coincide if and only if their dual configurations \(Z\) do not lie on the set of nonsingular points of a stable normal rational curve of degree \(n\) (a Torelli type theorem). In fact, the conjecture has been verified for arrangements which are normal crossing divisors of degree at least \(n+2\) and for arrangements consisting of up to 6 lines.
The aim of the paper under review is to prove a necessary and sufficient condition in terms of the dual configurations \(Z\) under which the corresponding arrangements with isomorphic sheaves \(\widetilde{\Omega}_{\mathbb P^n}(\log D_Z)\) do not coincide. The key idea of the proof is a functorial definition of \(\widetilde{\Omega}_{\mathbb P^n}(\log D_Z)\) as the dualized direct image of the sheaf of linear forms vanishing at \(Z\) in \(\mathbb P^n.\) As an application, the authors describe configurations satisfying this condition on stable rational curves and obtain counterexamples to the “only if” part of the conjecture. They demonstrate also that this part of the conjecture holds on the projective plane \(\mathbb P^2,\) even without the assumption on semi-stability of \(\widetilde{\Omega}_{\mathbb P^2}(\log D_Z).\) Above all, in the projective space \(\mathbb P^3\) some exceptions to the “if” part of the conjecture are considered, and so on.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C34 Torelli problem
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
32S22 Relations with arrangements of hyperplanes
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References:

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