Length and decomposition of the cohomology of the complement to a hyperplane arrangement

Bøgvad, Rikard; Gonçalves, Iara

doi:10.1090/proc/14379

Length and decomposition of the cohomology of the complement to a hyperplane arrangement
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by Rikard Bøgvad and Iara Gonçalves PDF

Proc. Amer. Math. Soc. 147 (2019), 2265-2273 Request permission

Abstract:

Let $\mathcal{A}$ 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 $Rj_*\mathbb {C}_{\tilde U}[n]$ 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 $Rj_*\mathbb {C}_{\tilde U}[n]$ as certain local cohomology sheaves and give their multiplicity. These results are implicitly contained, with different proofs, in Looijenga [Contemp. Math., 150 (1993), pp. 205–228], Budur and Saito [Math. Ann., 347 (2010), no. 3, 545–579], Petersen [Geom. Topol., 21 (2017), no. 4, 2527–2555], and Oaku [Length and multiplicity of the local cohomology with support in a hyperplane arrangement, arXiv:1509.01813v1].

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