Lifting Chern classes by means of Ekedahl–Oort strata

Abstract

The moduli space ${\mathcal{𝒜}}_g$ of principally polarized abelian varieties of genus $g$ is defined over $\mathbb{Z}$ and admits a minimal compactification ${\mathcal{𝒜}}_g^{\ast }$, also defined over $\mathbb{Z}$. The Hodge bundle over ${\mathcal{𝒜}}_g$ has its Chern classes in the Chow ring of ${\mathcal{𝒜}}_g$ with $\mathbb{Q}$-coefficients. We show that over ${\mathbb{𝔽}}_p$, these Chern classes naturally lift to ${\mathcal{𝒜}}_g^{\ast }$ and do so in the best possible way: despite the highly singular nature of ${\mathcal{𝒜}}_g^{\ast }$ they are represented by algebraic cycles on ${\mathcal{𝒜}}_g^{\ast }\otimes {\mathbb{𝔽}}_p$ which define elements in the bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky–Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.