G-zips and Ekedahl-Oort strata for Hodge type Shimura varieties

let (G, X ) be a Shimura datum of Hodge type, and Ebe its reflex field. Let p > 2 be a prime such that (G, X ) has good reduction. Let v be a place of E over p . Let Kp ⊂ G(Qp) be a hyperspecial subgroup, and K p ⊂ G(Ap ) be a compact open subgroup which is small enough. Let K = KpK p. By works of Deligne, we know that the smooth complex variety ShK (G, X )(C) := G(Q)\(X × G(Af )/K ) has a canonical model ShK (G, X ) over E. By recent works of Vasiu and Kisin, the E-variety has an integral canonical model SK (G, X ) over OE,v . The scheme SK (G, X ) is smooth over OE,v and uniquely determined by a certain extension property. Let kv = OE,v /(v) and S0 be the special fiber of SK (G, X ). The goal of this paper is to develop a theory of Ekedahl-Oort stratification for S0, generalizing known theory for PEL Shimura varieties developed by Oort, Moonen, Wedhorn, Viehmann... Thanks to works of Pink, Wedhorn and Ziegler on G -zips, we have the definition and technical tools for such a theory. Fixing a symplectic embedding, our first main result is the construction of a G-zip over S0 . This induces a morphism ξ : S0 → G − Zipµ , where G − Zipµ is the stack of G-zips of type µ constructed by Pink, Wedhorn and Ziegler. Fibers of ξ are defined to be Ekedahl-Oort strata. Our second main result is that ξ is smooth. One can then transfer knowledge about geometry of G − Zipµ to results about Ekedahl-Oort strata. In particular, we have a dimension formula for each non-empty stratum, and we know which strata lie in the closure of a given stratum.