Sections of the Hodge bundle over Ekedahl-Oort strata of Shimura varieties of Hodge type

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. We also deduce the fact that the $\mu$-ordinary locus is determined by the Ekedahl-Oort strata of its image by any embedding.