Noether-Lefschetz cycles on the moduli space of abelian varieties
Aitor Iribar Lopez
TL;DR
The paper develops an intersection-theoretic framework for Noether-Lefschetz cycles on the moduli of abelian varieties, introducing a tautological projection and proving it acts as a homomorphism on NL-supported classes. It combines Hecke correspondences and Torelli pullbacks to show non-tautologicality of certain product cycles in high genus, and it connects NL cycles to Gromov-Witten theory via the universal elliptic curve. The authors derive explicit degree formulas for polarization-morphism maps $\phi_{\delta}$ and $\pi_{\delta}$, compute the volume of Hecke correspondences, and propose modularity-type conjectures linking NL cycles to modular forms; together these results illuminate the structure of NL cycles, their interactions under Hecke operators, and their GW-theoretic significance. The work advances understanding of tautological versus non-tautological components in $\mathsf{CH}^*(\mathcal{A}_g)$ and lays groundwork for further connections to modularity and quantum cohomology.
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.