Abstract
We address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with the same monodromy around the planes.
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Notes
Note that this gives a positive answer to Question 3.5 in dimension three case.
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We thank Botong Wang for his help with various questions related to this paper. We thank the referees for helping us to improve the paper.
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Dedicated to the memory of Prof. Ĺžtefan Papadima.
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The first author was partly supported by the grants STRT/13/005 and Methusalem METH/15/026 from KU Leuven, and G0B2115N, G097819N, and G0F4216N from the Research Foundation — Flanders. The second author was supported by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.
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Budur, N., Liu, Y. On the length of perverse sheaves on hyperplane arrangements. European Journal of Mathematics 6, 681–712 (2020). https://doi.org/10.1007/s40879-019-00371-2
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DOI: https://doi.org/10.1007/s40879-019-00371-2