Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Decomposition of perverse sheaves
Stockholm University, Faculty of Science, Department of Mathematics.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2018. , p. 38
Keywords [en]
Sheaf cohomology, hyperplane arrangements, D-modules
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-155464ISBN: 978-91-7797-308-9 (print)ISBN: 978-91-7797-309-6 (electronic)OAI: oai:DiVA.org:su-155464DiVA, id: diva2:1200324
Public defence
2018-06-14, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Sida - Swedish International Development Cooperation Agency
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript.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

Available from: 2018-05-22 Created: 2018-04-24 Last updated: 2018-05-23Bibliographically approved
List of papers
1. Decomposition of perverse sheaves on plane line arrangements
Open this publication in new window or tab >>Decomposition of perverse sheaves on plane line arrangements
2018 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 6, p. 2476-2487Article in journal (Refereed) 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.

Keywords
Hyperplane arrangements, intersection cohomology
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-155422 (URN)10.1080/00927872.2017.1399410 (DOI)000428807400015 ()
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2018-04-20 Created: 2018-04-20 Last updated: 2018-04-30Bibliographically approved
2. Length and decomposition of the cohomology of the complement to a hyperplane arrangement
Open this publication in new window or tab >>Length and decomposition of the cohomology of the complement to a hyperplane arrangement
(English)Manuscript (preprint) (Other academic)
Keywords
sheaf cohomology, hyperplane arrangements
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-155425 (URN)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2018-04-20 Created: 2018-04-20 Last updated: 2019-01-28
3. Support of decomposition factors of direct images of line bundles on the open complement of a hyperplane configuration
Open this publication in new window or tab >>Support of decomposition factors of direct images of line bundles on the open complement of a hyperplane configuration
(English)Manuscript (preprint) (Other academic)
Keywords
sheaf cohomology, hyperplane arrangements, D-modules
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-155427 (URN)
Funder
Sida - Swedish International Development Cooperation Agency
Available from: 2018-04-20 Created: 2018-04-20 Last updated: 2018-05-18Bibliographically approved

Open Access in DiVA

Decomposition of perverse sheaves(268 kB)159 downloads
File information
File name FULLTEXT02.pdfFile size 268 kBChecksum SHA-512
f0ad16e21bdafb4a4ca83ae8a5fc316faef5b29ab2666fb778a3d047aaa87d8626470f209eba93e16bfb507f6033a1f2194fa70912778b3baa2a07b27df998a8
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Gonçalves, Iara Cristina Alvarinho
By organisation
Department of Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 159 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 571 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf