Foliations on Shimura varieties in positive characteristic

This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties S S in characteristic p p . The first, which we call tautological foliations, are defined on Hilbert modular varieties, and lift to characteristic 0 0 . The second, the V V -foliations, are defined on unitary Shimura varieties in characteristic p 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 p -closed, and the locus where they are smooth. Where not smooth, we construct a successive blowup 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 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 p , to S . S.