Lifting Chern classes by means of Ekedahl-Oort strata

TL;DR

This paper demonstrates that Chern classes of the Hodge bundle on the moduli space of abelian varieties can be algebraically lifted over finite fields, revealing new integral cycle representatives despite the space's singularities.

Contribution

It establishes algebraic lifts of Chern classes to the minimal compactification over finite fields, contrasting with their complex cohomology counterparts.

Findings

Chern classes lift to algebraic cycles over F_p

Lifts exist despite singularities of the compactification

Contrasts with nontrivial Tate extensions in complex cohomology

Abstract

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its 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.