HodgeChern classes and strataeffectivity 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 WedhornZiegler tautological rings $T^*(X), T^*(Y)$ of $X$ and its flag space $Y \to GZipFlag^{\mu}$ are \textit{strataeffective}, meaning that they are nonnegative 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 $\T^*(X)=\text{CH}_{\mathbf{Q}}(GZip^{\mu})$, $T^*(Y)=\text{CH}_{\mathbf{Q}}(GZipFlag^{\mu})$ are the entire Chow ring, and (2) When $X$ is the special fiber of an integral canonical model of a Hodgetype Shimura variety  in this case the strata are also known as EkedahlOort strata. We focus on the strataeffectivity of three types of classes: (a) Effective tautological classes, (b) Chern classes of GriffithsHodge bundles and (c) Generically $w$ordinary curves. We connect the question of strataeffectivity in (a) to the global section `Cone Conjecture' of GoldringKoskivirta. For every representation $r$ of $G$, we conjecture that the Chern classes of the GriffithsHodge bundle associated to $(G, \mu,r)$ are all strataeffective. This provides a vast generalization of a result of Ekedahlvan 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
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.05727
 arXiv:
 arXiv:2404.05727
 Bibcode:
 2024arXiv240405727C
 Keywords:

 Mathematics  Algebraic Geometry