In recent years, the theory of perverse sheaves has become a relevant tool for studying the topology of singular spaces. The aim of this course is to give an introduction to the theory perverse sheaves and discuss important aplications of this theory, as the Decompostion Theorem. Attendees are expected to be familiar with the theory of sheaves, and it is recommended to be familiar with the derived category.
Description
The duration of the course will be 10 hours. From January 20-24, 16:30-18:30.
Location: UPV/EHU Mathematics Department Seminar. (Campus Leioa)
The following topics will be covered in five lectures, tentatively distributed as follows, which may be modified depending on the time and pace of the lectures.
(1) Preliminaries and motivation.
- Derived category and derived functors.
- Deligne’s intersection complex.
(2) The perverse t-structure.
- Definition and properties of t-structures.
- Examples: the standard t-structure and the perverse t-structure.
- Gluing t-structures.
(3) The intermediate extension and simple objects.
- Definition of intermediate extension.
- Simple objects in the category of perverse sheaves.
(4) The decomposition package.
- The decomposition theorem.
- Hard-Lefschetz theorem.
- The semi-simplicity theorem.
(5) Nerby and vanishing cycles.
- Construction.
- Relation with perverse sheaves.
References
[1] Banagl, M. Topological Invariants of Stratified Spaces. Springer Science & Business Media, 2007.
[2] de Cataldo, M. A. A., and Migliorini, L. The Hodge Theory of Algebraic Maps. Ann. Sci. Ec. Norm. Supér., 38 (5), 693-750, 2005.
[3] Goresky, M., and MacPherson, R. Intersection homology II. Invent. Math., 72 (1), 77-129, 1983.
[4] Maxim, L. Intersection Homology & Perverse Sheaves with Applications to Singularities Vol. 281. Graduate Texts in Mathematics. Springer, Cham, pp. xv+270, 2019.