Hodge-Chern classes and strata-effectivity in tautological rings

Given a connected, reductive ${\mathbf{F}}_p$-group $G$, a cocharacter $\mu \in X_{\ast }(G)$ and a smooth zip period map ζ:X \to \mathop{\text{G-{\tt Zip}}}\nolimits^μ, we study which classes in the Wedhorn-Ziegler tautological rings $T^{\ast }(X),T^{\ast }(Y)$ of $X$ and its flag space $Y\to G-ZipFla{g}^{\mu }$ are $\mathit{\text{strata-effective}}$, meaning that they are non-negative rational linear combinations of pullbacks of classes of zip (flag) strata closures. Two special cases are: (1) When $X=G{\text{-Zip}}^{\mu }$ and the tautological rings ${\text{T}}^{\ast }(X)={\text{CH}}_{\mathbf{Q}}(G-Zi{p}^{\mu })$, $T^{\ast }(Y)={\text{CH}}_{\mathbf{Q}}(G-ZipFla{g}^{\mu })$ are the entire Chow ring, and (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 © 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 $(G,\mu ,r)$ 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 ${\text{Acal}}_{g,{\mathbf{F}}_p}$ in characteristic $p$ are represented by the closures of $p$-rank strata. We prove several instances of our conjecture