Foliations on unitary Shimura varieties in positive characteristic

When is inert in the quadratic imaginary field and <![CDATA[$m, unitary Shimura varieties of signature and a hyperspecial level subgroup at , carry a natural foliation of height 1 and rank in the tangent bundle of their special fiber . 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 , a successive blow-up of . Over the ( -)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber of a certain Shimura variety with parahoric level structure at . As a result, we get that this 'horizontal component' of , 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 , and a certain Ekedahl-Oort stratum that we denote . We conjecture that these are the only integral submanifolds.