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Length and decomposition of the cohomology of the complement to a hyperplane arrangement. (English) Zbl 1442.55003

Summary: Let \(\mathscr A\) be a hyperplane arrangement in \( \mathbb{C}^n\). We prove in an elementary way that the number of decomposition factors as a perverse sheaf of the direct image \( Rj_\ast\mathbb{C}_{\tilde U}[n]\) of the constant sheaf on the complement \( {\tilde U}\) to the arrangement is given by the Poincaré polynomial of the arrangement. Furthermore, we describe the decomposition factors of \( Rj_\ast\mathbb{C}_{\tilde U}[n]\) as certain local cohomology sheaves and give their multiplicity. These results are implicitly contained, with different proofs, in [E. Looijenga, Contemp. Math. 150, 205–228 (1993; Zbl 0814.14029); N. Budur and M. Saito, Math. Ann. 347, No. 3, 545–579 (2010; Zbl 1195.14070); D. Petersen, Geom. Topol. 21, No. 4, 2527–2555 (2017; Zbl 1420.55027); O. T. Oaku, “Length and multiplicity of the local cohomology with support in a hyperplane arrangement”, Preprint, arXiv:1509.01813].

MSC:

55N30 Sheaf cohomology in algebraic topology
32C38 Sheaves of differential operators and their modules, \(D\)-modules
32S22 Relations with arrangements of hyperplanes
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References:

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