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 that these singularities are resolved if we pass to a natural smooth moduli problem $S^{\sharp }$ , a successive blow-up of $S$ . Over the ( $\unicode[STIX]{x1D707}$ -)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-étale locus in the special fiber $S_{0}(p)$ 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 $S_{\text{fol}}$ . We conjecture that these are the only integral submanifolds.