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].References
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Additional Information
- Rikard Bøgvad
- Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
- Email: rikard@math.su.se
- Iara Gonçalves
- Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
- Email: iaragoncalves@uem.mz; iaraalvgon@gmail.com
- Received by editor(s): January 2, 2018
- Received by editor(s) in revised form: September 3, 2018
- Published electronically: January 28, 2019
- Additional Notes: The second author gratefully acknowledges financing by SIDA/ISP
- Communicated by: Lev Borisov
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2265-2273
- MSC (2010): Primary 55N30; Secondary 32C38
- DOI: https://doi.org/10.1090/proc/14379
- MathSciNet review: 3937700