Hodge-Chern classes and strata-effectivity in tautological rings

Abstract

Given a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and a smooth zip period map $\zeta:X \to \mathop{\text{$G$-{\tt Zip}}}\nolimits^{\mu}$, we study which classes in the Wedhorn-Ziegler tautological rings $T^*(X), T^*(Y)$ of $X$ and its flag space $Y \to G-ZipFlag^{\mu}$ are \textit{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 $\T^*(X)=\text{CH}_{\mathbf{Q}}(G-Zip^{\mu})$, $T^*(Y)=\text{CH}_{\mathbf{Q}}(G-ZipFlag^{\mu})$ 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 (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 $(G, \mu,r)$ 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 $\Acal_{g,\mathbf{F}_p}$ in characteristic $p$ are represented by the closures of $p$-rank strata. We prove several instances of our conjecture