TY - JOUR

T1 - Length and decomposition of the cohomology of the complement to a hyperplane arrangement

AU - Bøgvad, Rikard

AU - Gonçalves, Iara

AU - Borisov, Lev

N1 - Funding Information:
We want to thank Rolf Källström for discussions on these topics, as well as Nero Budur and Dan Petersen for directing us to [4], respectively, [11] and [14]. The second author gratefully acknowledges financing by SIDA/ISP.

PY - 2019/5

Y1 - 2019/5

N2 - Let A be a hyperplane arrangement in ℂ n . We prove in an elementary way that the number of decomposition factors as a perverse sheaf of the direct image Rj*ℂŨ [n] of the constant sheaf on the complement Ũ to the arrangement is given by the Poincaré polynomial of the arrangement. Furthermore, we describe the decomposition factors of Rj*ℂŨ [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].

AB - Let A be a hyperplane arrangement in ℂ n . We prove in an elementary way that the number of decomposition factors as a perverse sheaf of the direct image Rj*ℂŨ [n] of the constant sheaf on the complement Ũ to the arrangement is given by the Poincaré polynomial of the arrangement. Furthermore, we describe the decomposition factors of Rj*ℂŨ [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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U2 - 10.1090/proc/14379

DO - 10.1090/proc/14379

M3 - Article

AN - SCOPUS:85065466584

VL - 147

SP - 2265

EP - 2273

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -