In this note we compute the tautological ring of non-compactified Hilbert modular varieties at an unramified prime. This is the first computation of the tautological ring of a non-compactified Shimura variety beyond the case of the Siegel modular variety. While the method generalises van der Geer's approach for the Siegel modular variety, there is an added difficulty in that the highest degree socle has d > 1 generators rather than 1. To deal with this we prove that the d cycle classes of codimension one Ekedahl-Oort strata closures are linearly independent. In contrast, in the Siegel modular case it suffices to prove that the cycle class of the p-rank zero locus is non-zero. The limitations of our method for computing the tautological ring of other non-compactified Shimura varieties are demonstrated with an instructive example.

Given a reductive group G over an algebraically closed field and subsets I,J ⊂ Δ of the simple roots Δ determined by a choice of maximal torus and Borel subgroup, there is a closed embedding of flag varieties L_J /L_J ∩ P_I ↪ G/P_I. In this paper we compute the class of the sub flag variety [L_J/L_J ∩ P_I] ∈ A^•(G/P_I) in the Chow ring for the ‘Siegel’ case where G is a general symplectic group of semisimple rank g and P_I is the parabolic stabilising a maximal isotropic subspace. This corresponds, under the isomorphism with the tautological ring of the moduli space of principally polarised abelian varieties , to the generator of the classes in the tautological ring which are supported on the toroidal boundary. This provides basic evidence for a conjecture describing the tautological ring of a Hodge-type Shimura variety.

Given a connected, reductive group G over the finite field F of order p, a cocharacter μ of G over an algebraic closure of F and a stack X admitting a 'zip period morphism' to the stack of G-Zips of type μ, we study which classes in the Wedhorn-Ziegler tautological rings of X and its flag space Y are strata-effective, meaning that they are non-negative rational linear combinations of pullbacks of classes of zip (flag) strata closures. Two special cases are: (1) When X is the stack of G-Zips and the tautological rings coincide with the entire Chow rings (2) When X is the special fiber of an integral canonical model of a Hodge-type Shimura variety – in this case the strata are also known as Ekedahl-Oort strata. We focus on the strata-effectivity of three types of classes: (a) Effective tautological classes, (b) Chern classes of Griffiths-Hodge bundles and (c) Generically w-ordinary curves. We connect the question of strata-effectivity in (a) to the global section ‘Cone Conjecture’ of Goldring-Koskivirta. For every representation r of G, we conjecture that the Chern classes of the Griffiths-Hodge bundle associated to (G, μ, r) are all strata-effective. This provides a vast generalization of a result of Ekedahl-van der Geer that the Chern classes of the Hodge vector bundle on the moduli space of principally polarized abelian varieties in characteristic p are represented by the closures of p-rank strata. We prove several instances of our conjecture, including the case of Hilbert modular varieties, where the conjecture says that all monomials in the first Chern classes of the factors of the Hodge vector bundle are strata-effective. We prove results about each of (a), (b) and (c) which have applications to Shimura varieties and also in cases where no Shimura variety exists.

Given a connected reductive group G over the finite field F of order p and a cocharacter μ of G over an algebraic closure of F, we compute the Grothendieck group of the stack of G-Zips of type μ (as a ring), under the additional assumption that the derived group of G is simply connected.