Canonical sections of the Hodge bundle over Ekedahl–Oort strata of Shimura varieties of Hodge type

Abstract

We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl–Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl–Oort strata. In the projective case, this gives a very short proof of non-emptiness of Ekedahl–Oort strata. It follows that the Newton strata are also nonempty, by a result of S. Nie. From the canonicity of our construction, we deduce the fact that the μ-ordinary locus is determined by the Ekedahl–Oort strata of its image under any embedding of Shimura varieties.