Purity of Zip Strata with Applications to Ekedahl-Oort Strata of Shimura   Varieties of Hodge Type
  We show that the zip stratification given by an arbitrary $\Ghat$-zip over a scheme is pure. We deduce purity of the level-$m$-stratification for truncated Barsotti-Tate groups and purity of the Ekedahl-Oort stratification for special fibers of good models of Shimura varieties of Hodge type. We also show that all Ekedahl-Oort strata are quasi-affine schemes and generalize several properties of the Bruhat stratification from the PEL case to the case of Shimura varieties of Hodge type. 
Torsten Wedhorn and Yaroslav Yatsyshyn
math.AG math.NT math.RT