Ekedahl-Oort stratifications of Shimura varieties via Breuil-Kisin windows

TL;DR

This paper constructs a morphism from the special fiber of a Hodge type Shimura variety to a quotient sheaf of the loop group, providing a new perspective on Ekedahl-Oort stratifications via Breuil-Kisin windows.

Contribution

It introduces a novel method to interpret Ekedahl-Oort strata using Breuil-Kisin windows and loop group invariants, linking deformation theory with geometric stratifications.

Findings

Constructed a morphism from the special fiber to a loop group quotient sheaf.

Reproduced Ekedahl-Oort strata as fibers of this morphism.

Provided a conceptual interpretation of Viehmann's invariants.

Abstract

Let $ S $ be the special fibre of the good reduction of a Shimura variety of Hodge type. By constructing adapted deformations for the associated $p$-divisible groups of $ S $, we manage to construct a morphism from $S$ to some quotient sheaf of the loop group associated with $S$. We show that the geometric fibres of this morphism give back the Ekedahl-Oort strata of $S$. For any geometric point $ x $ of $ S $, we give a deformation over $ W(k(x)) $ of the $ p $-divisible group associated with $ x $ by (non-canonically) constructing a Breuil-Kisin window (which corresponds to a $ p $-divisible group over $ W(k(x)) $ by the work of Kisin). This map in a sense gives a conceptual interpretation of Viehmann's new invariants "truncations of level one of elements in the loop group".