Iara Goncalves: Decomposition of perverse sheaves

Abstract:
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.