Cours fondamental 2 (GA, TA)
Perverse sheaves and decomposition theorem
Mirko Mauri
Contact : mauri à imj-prg.fr
Des notes de cours seront disponibles.
Langue du cours : English
Présentation
Perverse sheaves and the decomposition theorem are essential tools in algebraic geometry, representation theory, and arithmetic, and they continue to drive new advancements and discoveries in pure mathematics. The decomposition theorem in particular is a powerful tool for investigating the topology of proper morphisms. In brief, it asserts that a proper morphism \(f\colon X \to Y\) of algebraic varieties decomposes the cohomology of \(X\) in fundamental building blocks that are the cohomology groups of perverse sheaves on \(Y\). The course will serve as an introduction to these concepts, from the viewpoint of Hodge theory and classical algebraic geometry.
The course will continue with the specialised course Decomposition theorem for Lagrangian fibrations.
Contenu
- Fibrations, monodromy, local systems, Leray spectral sequence
- Intersection cohomology and perverse sheaves
- Cohomology and derived category of sheaves
- Decomposition theorem after de Cataldo and Migliorini
Prérequis
Géométrie complexe et théorie de Hodge; Schémas I; Topologie algébrique des variétés I. While attending the course, you are warmly invited to follow Schémas II. Next term, you may consider to attend also the course Champs algébriques
Bibliographie
- C. Voisin. Hodge Theory and Complex Algebraic Geometry II. (Section 1 and 2)
- R. Bott, L.W. Tu. Differential Forms in Algebraic Topology. (Section 14)
- A. Beilinson, J. Bernstein, P. Deligne. Faisceaux pervers.
- M.A.A. de Cataldo. Perverse sheaves and the topology of algebraic varieties .
- M.A.A. de Cataldo, L. Migliorini. The Hodge theory of algebraic maps.
- M.A.A. de Cataldo, L. Migliorini. The Decomposition Theorem and the topology of algebraic maps.