Hodge stratification in low dimension

TL;DR

This paper investigates the Hodge stratification of the special fiber of certain Shimura varieties with Pappas-Rapoport conditions, focusing on low ramification indices ($e \,\leq\, 3$), revealing detailed stratification structures.

Contribution

It introduces a detailed analysis of Hodge stratification for low ramification cases, including multiple polygons for $e=3$, enhancing understanding of the sheaf of differentials with extra structure.

Findings

For $e \,\leq\, 2$, the Hodge polygon induces a strong stratification.

For $e=3$, multiple polygons are needed to describe the stratification.

The stratification describes the isomorphism class of the sheaf of differentials with extra structure.

Abstract

We define and study the Hodge stratification for the special fiber of Shimura varieties defined with the Pappas-Rapoport condition, in the case of low ramification index ($e \leq 3$). For $e \leq 2$, the Hodge polygon induces a strong stratification. For $e=3$, one needs to introduce several polygons. They describe the isomorphism class of the sheaf of differentials with extra structure, and induce a strong stratification on the variety.