The zero section of the universal semiabelian variety, and the double ramification cycle

TL;DR

This paper computes the class of the partial compactification of the universal zero section in the Chow ring of the boundary of the universal family of ppav, extending previous results and contributing to the understanding of Chow groups in toroidal compactifications.

Contribution

It provides a formula for the class of the partial compactification of the universal zero section, extending known results and aiding the study of Chow groups of toroidal compactifications.

Findings

Derived the class of the partial compactification of the zero section

Extended Hain's results on the double ramification cycle to Jacobians of curves

Provided a framework for inductive study of Chow groups in toroidal compactifications

Abstract

We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of the moduli space of ppav. Our formula provides a first step in a program to understand the Chow groups of toroidal compactifications of the moduli of ppav, especially of the perfect cone compactification, by induction on genus. By restricting to the locus of Jacobians of curves, our results extend the results of Hain on the double ramification (two-branch-point) cycle.