Foliations on Shimura varieties in positive characteristic

TL;DR

This paper investigates two types of foliations on Shimura varieties in characteristic p, analyzing their properties, smoothness, and relations to other Shimura varieties, extending previous work on foliations in this context.

Contribution

It introduces and studies tautological and V-foliations on Shimura varieties in characteristic p, including criteria for p-closure, smoothness, and their geometric and algebraic implications.

Findings

Determined when foliations are p-closed and smooth.

Constructed blow-ups to extend foliations as smooth.

Related foliations to purely inseparable maps between Shimura varieties.

Abstract

This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the "$V$-foliations", are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a "successive blow up" of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of $S$ by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at $p$, to $S.$