The supersingular locus of the Shimura variety of $\mathrm{GU}(2,n-2)$

TL;DR

This paper investigates the structure of the supersingular locus in a specific Shimura variety, describing its irreducible components and their intersections using flag schemes, Deligne--Lusztig varieties, and Frobenius stratifications.

Contribution

It provides an explicit geometric description of the supersingular locus of the (2,n-2) Shimura variety, including component intersections, via flag schemes and Deligne--Lusztig varieties.

Findings

Irreducible components realized as closed subschemes of flag schemes

Explicit conditions for components using Deligne--Lusztig varieties

Stratifications of intersections via Frobenius powers

Abstract

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to $\mathrm{GU}(2,n-2)$. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne--Lusztig varieties defined by explicit conditions after taking perfections. Moreover we study the intersections of the irreducible components. Stratifications of Deligne--Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections.