Tautological projection for cycles on the moduli space of abelian varieties
Samir Canning, Sam Molcho, Dragos Oprea, Rahul Pandharipande
TL;DR
The paper defines a canonical tautological projection on the moduli space $ ext{A}_g$ of principally polarized abelian varieties, decomposing any cycle into tautological and non-tautological parts. A central technical achievement is proving that the top Hodge-class $\lambda_g$ vanishes on the boundary of every toroidal compactification, via a boundary-residue analysis in logarithmic geometry, enabling a compactification-independent projection and a well-defined $\lambda_g$-pairing. The authors compute the projections of product cycles through a determinant formula tied to Schubert calculus on the Lagrangian Grassmannian, relate tautological rings across $ar{ ext{A}}_g$ and $ ext{LG}_g$, and establish explicit results for product loci, with applications to Torelli/Noether–Lefschetz-type questions and a conjectural appendix on real multiplication loci. The work blends log geometry, semistable degeneration, and Schubert calculus to provide structural insights into cycles on $ ext{A}_g$ and its toroidal compactifications, offering a robust framework for further exploration of tautological versus non-tautological components. The Appendix conjectures about real multiplication loci hint at deeper connections between boundary geometry and arithmetic Noether–Lefschetz phenomena.
Abstract
We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space $\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the boundary geometry. The existence of the projection operator raises many natural questions about cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles $\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication.