On the length of perverse sheaves on hyperplane arrangements
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On the length of perverse sheaves on hyperplane arrangements Nero Budur1
· Yongqiang Liu2,3
Dedicated to the memory of Prof. S¸ tefan Papadima Received: 5 March 2019 / Revised: 2 September 2019 / Accepted: 4 September 2019 © Springer Nature Switzerland AG 2019
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. Keywords Local system · Perverse sheaf · D-module · Riemann–Hilbert correspondence Mathematics Subject Classification 14F05 · 14F10 · 14F45 · 32C38
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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Nero Budur [email protected] Yongqiang Liu [email protected]
1
Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
2
Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
3
Present Address: The Institute of Geometry and Physics, University of Science and Technology of China, No. 96, JinZhai Road Baohe District, Hefei 230026, Anhui, People’s Republic of China
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N. Budur, Y. Liu
1 Introduction 1.1 Overview Perverse sheaves and intersection cohomology are fundamental objects encoding the complexity of the topology of a stratified space. Every perverse sheaf on a complex algebraic variety (and using field coefficients for sheaves) has a finite maximal filtration, called the composition series, with non-zero simple successive quotients. The length of a perverse sheaf counts the number of simple objects in any composition series. There is currently no known algorithm to compute the length of a perverse sheaf. The simplest perverse sheaves are the local systems. Understanding the length of local systems amounts to understanding the geometry of the GIT quotient map from the space of representations of the fundamental group of the variety to the moduli of (semi-simple) local systems. A next natural class of perverse sheaves to consider are direct images via the open embedding of local systems on the complement of a hypersurface. In this article we address the length of such perverse sheaves on Cn constructed from rank one local systems on the complement of an arrangement of hyperplanes. 1.2 Notation We denote by Perv(Cn ) the category of C-perverse sheaves on Cn. Let G(Cn ) be the Grothendieck group
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