Exemples de classification du champ des faisceaux pervers

TL;DR

This thesis employs stack theory to describe and classify perverse sheaves on stratified spaces, providing new descriptions and equivalences especially for spaces stratified by normal crossings and hyperplane arrangements.

Contribution

It introduces a stack-theoretic framework to analyze perverse sheaves, extending existing descriptions to more complex stratified spaces like toric varieties and hyperplane arrangements.

Findings

New stack-based description of perverse sheaves on stratified spaces

Equivalence between stack sections and quiver representations

Application to locally toric and hyperplane stratified spaces

Abstract

In this thesis we show how to use stack theory to glue description of the category of perverse sheaves P(X,S) on a stratified space (X,S). Hence we give new description of P(X,S) when X is locally C^n stratified by the stratification S given by the normal crossing. First we give a characterization of the 2-category of a stack on a stratified space. Thank to this and to a description in term of quiver's representation of the category P(C^n,S) due to Galligo, Granger, Maisonobe, we define a stack on C^n constructible relatively to S, equivalent to the stack Perv(C,S) of perverse sheaves on C^n relatively to S. As a stack can be define on an open covering and as toric varieties and C^2 stratified by a generic hyperplanes arrangement are locally isomorphic to (C^n,S) we can define a stack CC on these spaces equivalent to the stack PervX. Then a study of the global sections of CC gives a category of quivers representations equivalent to the category P(X,S').