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  • 1.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Decomposition of perverse sheaves on plane line arrangements2018In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 6, p. 2476-2487Article in journal (Refereed)
    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.

  • 2.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Length and decomposition of the cohomology of the complement to a hyperplane arrangementManuscript (preprint) (Other academic)
  • 3.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    LENGTH AND DECOMPOSITION OF THE COHOMOLOGY OF THE COMPLEMENT TO A HYPERPLANE ARRANGEMENT2019In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 147, no 5, p. 2265-2273Article in journal (Refereed)
    Abstract [en]

    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(*) C-(U) over tilde[n] of the constant sheaf on the complement (U) over tilde to the arrangement is given by the Poincare polynomial of the arrangement. Furthermore, we describe the decomposition factors of Rj(*) C-(U) over tilde[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].

  • 4.
    Bøgvad, Rikard
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gonçalves, Iara
    Stockholm University, Faculty of Science, Department of Mathematics.
    Support of decomposition factors of direct images of line bundles on the open complement of a hyperplane configurationManuscript (preprint) (Other academic)
  • 5.
    Gonçalves, Iara Cristina Alvarinho
    Stockholm University, Faculty of Science, Department of Mathematics.
    Decomposition of perverse sheaves2018Doctoral 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.

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