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Decomposition of perverse sheaves on plane line arrangements
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.ORCID-id: 0000-0001-9439-2276
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
2018 (Engelska)Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, nr 6, s. 2476-2487Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

On the complement X = C-2 - U-i=1(n) L-i to a central plane line arrangement U-i=1(n) L-i subset of C-2, a locally constant sheaf of complex vector spaces L-a is associated to any multi-index aC(n). Using the description of MacPherson and Vilonen of the category of perverse sheaves [7, 8], we obtain a criterion for the irreducibility and number of decomposition factors of the direct image j : x -> C-2 as a perverse sheaf, where j:XC2 is the canonical inclusion.

Ort, förlag, år, upplaga, sidor
2018. Vol. 46, nr 6, s. 2476-2487
Nyckelord [en]
Hyperplane arrangements, intersection cohomology
Nationell ämneskategori
Matematik
Forskningsämne
matematik
Identifikatorer
URN: urn:nbn:se:su:diva-155422DOI: 10.1080/00927872.2017.1399410ISI: 000428807400015OAI: oai:DiVA.org:su-155422DiVA, id: diva2:1199349
Forskningsfinansiär
Sida - Styrelsen för internationellt utvecklingssamarbeteTillgänglig från: 2018-04-20 Skapad: 2018-04-20 Senast uppdaterad: 2018-04-30Bibliografiskt granskad
Ingår i avhandling
1. Decomposition of perverse sheaves
Öppna denna publikation i ny flik eller fönster >>Decomposition of perverse sheaves
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This PhD thesis consists in three papers in which we describe irreducibility conditions and the number of factors in a composition series of certain perverse sheaves. We study some particular cases, providing examples and showing how to explicitly use perverse sheaves to obtain precise results. The aim is to add to the class of concrete applications of perverse sheaves and exploit their role in the cohomology of hyperplane arrangements. In the three papers the perverse sheaves considered are given by the derived direct image of locally constant sheaves defined in the complement  U of a hyperplane arrangement. In Paper I, we start with a locally constant rank 1 sheaf on U and use a category equivalence, developed by MacPherson and Vilonen, to obtain a criterion for the irreducibility in terms of a multi-index that determines the locally constant sheaf. We then determine the number of decomposition factors when the irreducibility conditions are not satisfied. In Paper II we consider the constant sheaf on U, show that the number of decomposition factors of the direct image is given by the Poincaré polynomial of the hyperplane arrangement, and furthermore describe them as certain local cohomology sheaves and give their multiplicity. In Paper III, we use the Riemann-Hilbert correspondence and D-module calculations to determine a condition describing when the direct image of a locally constant sheaf contains a decomposition factor as a perverse sheaf that has support on a certain flat of the hyperplane arrangement.

Ort, förlag, år, upplaga, sidor
Stockholm: Department of Mathematics, Stockholm University, 2018. s. 38
Nyckelord
Sheaf cohomology, hyperplane arrangements, D-modules
Nationell ämneskategori
Matematik
Forskningsämne
matematik
Identifikatorer
urn:nbn:se:su:diva-155464 (URN)978-91-7797-308-9 (ISBN)978-91-7797-309-6 (ISBN)
Disputation
2018-06-14, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Sida - Styrelsen för internationellt utvecklingssamarbete
Anmärkning

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript.

Tillgänglig från: 2018-05-22 Skapad: 2018-04-24 Senast uppdaterad: 2018-05-23Bibliografiskt granskad

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