Cycles on the Moduli Space of Abelian Varieties

TL;DR

This paper studies algebraic cycles on the moduli space of abelian varieties, determining the tautological ring, torsion bounds, and cycle classes of stratifications, advancing understanding of geometric and arithmetic properties in characteristic p.

Contribution

It provides new results on the tautological ring, torsion bounds of the top Chern class, and cycle classes of Ekedahl-Oort stratification, including formulas for specific loci.

Findings

Tautological ring structure determined

Bounds established for torsion of mbda_g

Cycle classes of Ekedahl-Oort stratification computed

Abstract

In this paper a number of results on cycles on the moduli space of principally polarized abelian varieties is presented. Results include a determination of the tautological ring, bounds on the order of torsion of the top Chern class $\lambda_g$ and a determination of the cycle classes of the Ekedahl-Oort stratification in characteristic $p$. It includes as special cases formulas for the classes of loci like the $p$-rank $\leq f$ locus or the $a$-number $\geq a$-locus. The results on the tautological ring are my own work, the results on the torsion of $\lambda_g$ and on the cycle classes of the Ekedahl-Oort stratification are joint work with Torsten Ekedahl and some of the results on curves are joint work with Carel Faber.