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].
All Science Journal Classification (ASJC) codes
- Applied Mathematics