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

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 we 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 ${\mathcal{A}}_g$. Our formula provides a first step in a program to understand the Chow groups of ${\overline{\mathcal{A}}}_g$, especially of the perfect cone compactification, by induction on genus. By restricting to the image of ${\mathcal{M}}_g$ under the Torelli map, our results extend the results of Hain on the double ramification cycle, answering Eliashberg’s question.