Noether-Lefschetz cycles on the moduli space of abelian varieties

TL;DR

This paper investigates the intersection properties of Noether-Lefschetz cycles on the moduli space of abelian varieties, revealing non-tautological classes in certain genera and connecting these cycles to Gromov-Witten theory.

Contribution

It introduces a new homomorphism for cycles supported on non-simple abelian varieties and demonstrates non-tautological classes for specific genera, linking to Gromov-Witten theory.

Findings

Projection defines a homomorphism on cycles supported on the non-simple locus.

The class [A_1 × A_{g-1}] is not tautological for g=12 and g≥16 even.

Connections established between Noether-Lefschetz cycles and Gromov-Witten theory.

Abstract

The locus of non-simple abelian varieties in the moduli space of principally polarized abelian varieties gives rise to Noether-Lefschetz cycles. We study their intersection theoretic properties using the tautological projection constructed in [CMOP24], and show that projection defines a homomorphism when restricted to cycles supported on that locus. Using Hecke correspondences and the pullback by Torelli we prove that $[\mathcal {A}_1 \times \mathcal A_{g-1}]$ is not tautological in the sense of [vdG99] for $g=12$ and $g\geq 16$ even. We also explore the connections between Noether-Lefschetz cycles and the Gromov-Witten theory of a moving elliptic curve.