Foliations on Shimura varieties in positive characteristic

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 $S.$