Filtrations de stratification de quelques vari\'et\'es de Shimura simples

TL;DR

This paper introduces new stratification filtrations for perverse sheaves on schemes, extending existing concepts like weight and monodromy filtrations, with applications to simple unitary Shimura varieties and vanishing cycles.

Contribution

It defines and studies new filtrations of stratification for perverse sheaves applicable beyond traditional filtrations, with concrete applications to Shimura varieties.

Findings

New filtrations generalize weight and monodromy filtrations.

Applications to simple unitary Shimura varieties and vanishing cycles.

Simplification of key steps in the analysis of perverse sheaves.

Abstract

We define and study new filtrations called of stratification of a perverse sheaf on a scheme; beside the cases of the weight or monodromy filtrations, these filtrations are available whatever are the ring of coefficients. We illustrate these constructions in the geometric situation of the simple unitary Shimura varieties of Harris and Taylor's book for the perverse sheaves of Harris-Taylor and the complex of vanishing cycles, introduced and studied in my 2009 paper at inventiones. In the situation studied in loc. cit., we show how to use these filtrations to simplify the principal step of this paper; the cases of finite field or ring of integer of a local field will be studied in the next published paper.