Singularities in the Ekedahl--Oort stratification

TL;DR

This paper investigates the geometric properties of Ekedahl--Oort strata in Shimura varieties, providing criteria for normality, Cohen--Macaulayness, and smoothness, along with explicit invariants and cycle class formulas.

Contribution

It offers new combinatorial and numerical criteria for the geometric properties of EO strata, including explicit formulas and invariants, advancing understanding of their structure.

Findings

Criteria for normality and Cohen--Macaulayness of EO strata unions

Explicit smoothness conditions for one-dimensional EO-stratum closures

Construction of reduced strata Hasse invariants and cycle class formulas

Abstract

We give conceptual and combinatorial criteria for the normality and Cohen--Macaulayness of unions of Ekedahl--Oort strata in the special fiber of abelian type Shimura varieties. For unions of two strata, one of the two having codimension one in the closure of the other, we determine exactly when their union is smooth. We provide explicit numerical criteria for the smoothness of any one-dimensional EO-stratum closure of Shimura varieties. For groups of type $\mathsf{B}_n$, we describe the smooth and normal loci of all EO-strata closures. We construct reduced strata Hasse invariants of explicit weights on EO-strata of codimension at most $n-1$, showing that their Zariski closures are local complete intersections. We also provide a closed form formula for their cycle classes.