Abstract: Inseparable morphisms proved to be
an important tool for the study of algebraic
varieties in characteristic p. In particular,
Rudakov-Shafarevitch, Miyaoka and Ekedahl
have constructed a dictionary between
"height 1" foliations in the tangent bundle
and "height 1" purely inseparable quotients
of a non-singular variety in characteristic p.
In a joint work with Eyal Goren we use this
dictionary to study the special fiber S of a
unitary Shimura variety of signature (n,m),
m < n, at a prime p which is inert in the
underlying imaginary quadratic field. We
construct a natural foliation in the "ordinary"
part of S (classifying \mu-ordinary abelian
varieties), and extend it to the whole of S^#,
which is a certain "successive blow-up" of S
at deeper Ekedahl-Oort strata. We identify the
resulting height-1 quotient morphism as a map
to the Zariski closure of the ordinary-e'tale
component of the special fiber of an integral
model of a Shimura variety S_0(p) with
parahoric level structure at p. As a result we
can show that the closure of the ordinary-e'tale
component in S_0(p) is non-singular. We
discuss integral manifolds to the foliation,
and formulate a new conjecture
"of Andre' - Oort type".
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