The supersingular locus of the Shimura variety of GU(1,n-1) II

TL;DR

This paper completes the geometric description of the supersingular locus in certain unitary Shimura varieties at an inert prime, revealing its stratification, irreducible components, and their relations to Deligne-Lusztig varieties.

Contribution

It provides a detailed stratification and geometric analysis of the supersingular locus for GU(1,n-1) Shimura varieties at inert primes, extending previous results for small n.

Findings

Supersingular locus is a local complete intersection.

Irreducible components are isomorphic to Deligne-Lusztig varieties.

Stratification aligns with the Bruhat-Tits building structure.

Abstract

We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\GU(1,n-1)$ over $\QQ$ in the case that $p$ is inert. This was started by the first author in \cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of $\Nbf$. We show that the combinatorial behaviour of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the reduced irreducible components of $\Ncal$) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that the supersingular locus is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of the supersingular locus.