The cycle class of the supersingular locus of principally polarized abelian varieties

TL;DR

This paper derives a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties, linking it to Chern classes and explicitly determining a key factor for genus 4.

Contribution

It provides a new explicit formula for the cycle class of the supersingular locus, extending known results to genus 4.

Findings

Formula for the cycle class in the Chow ring

Determination of the polynomial factor for genus 4

Connection to Chern classes of the Hodge bundle

Abstract

We prove a formula for the cycle class of the supersingular locus in the Chow ring with rational coefficients of the moduli space of principally polarized abelian varieties in characteristic $p$. This formula determines this class as a monomial in the Chern classes of the Hodge bundle up to a factor that is a polynomial in $p$. This factor is known for $g\leq 3$. We determine the factor for $g=4$.