Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties

Abstract

We introduce a stratification on the space of symplectic flags on the de Rham bundle of the universal principally polarized abelian variety in positive characteristic and study its geometric properties like irreducibility of the strata and we calculate the cycle classes. When the characteristic p is treated as a formal variable these classes can be seen as a deformation of the classes of the Schubert varieties for the corresponding classical flag variety (the classical case is recovered by putting p equal to 0). We relate our stratification with the E-O stratification on the moduli space of principally polarized abelian varieties of a fixed dimension and derive properties of the latter. Our results are strongly linked with the combinatorics of the Weyl group of the symplectic group.

Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties

“…In Remark 2.5.7 we shall see that S w is contained in the supersingular locus if and only if w ∈ I W (c) g with c ≤ g/2 . Ekedahl and van der Geer proved in [8], Theorem 11.5 that S w is irreducible for every w ∈ I W (c) g with c > g/2 ; thus (i) was proved. In the last section we shall confirm (ii) by showing lim p→∞ H g,c = ∞.…”

“…The main reference for the EOstratification is [27]. For a formulation in terms of Weyl groups, see [8], [19] and [20]. Definition 2.5.1.…”

Supersingular Kottwitz–Rapoport strata and Deligne–Lusztig varieties

“…It is not clear to us whether this inclusion is an equality. We also mention the paper [12] by Ekedahl and van der Geer, in which the EO stratification of A g is investigated using a flag complex F g → A g , which contains part of A I . See [18] for some results about the relationship between the KR stratification and the Ekedahl-Oort stratification.…”

Automorphic Forms on the Stack of G-Zips

“…The function(5.3.3) is homogeneous, so its weight lies in C zip by definition. The case Sp(4). In this section we specialize the discussion to the case n = 2.…”