Foliations on unitary Shimura varieties in positive characteristic

When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($\mu$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of $S^\sharp$ by the foliation, and identify it as the Zariski closure of the ordinary-\'etale locus in the special fibre $S_0(p)$ of a certain Shimura variety with parahoric level structure at $p$. As a result we get that this "horizontal component" of $S_0(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen-Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$, and a certain Ekedahl-Oort stratum that we denote $S_{fol}$. We conjecture that these are the only integral submanifolds.