Noether-Lefschetz cycles on the moduli space of abelian varieties

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• Aitor Iribar Lopez, ETH Zurich
• Wednesday 21 May 2025, 14:15-15:15
• CMS MR13.

If you have a question about this talk, please contact Dhruv Ranganathan.

We study $A_g$, the moduli space of principally polarized abelian varieties of dimension $g$. The tautological ring, generated by the chern classes of the Hodge bundle, was fully determined by Gerard van der Geer in 1999, but the question of which geometrically defined cycles belong to this subring remains open. In 2024, Canning, Oprea and Pandharipande showed that $[A_1 \times A_5]$ is not tautological in $A_6$, and later I showed that $[A_1 \times A_{g-1}]$ is not tautological for $g=12$ or $g \geq 16$ even.

The cycle $[A_1\times A_{g-1}]$ is one of the Noether-Lefschetz cycles on the moduli spaces. With Greer and Lian, we conjecture that these cycles are related to modular forms of weight $2g$.

A new technique, which was not available in 1999 is the existence of a projection operator by Canning, Oprea, Molcho and Pandharipande onto the tautological ring. This leads to interesting conjectures about Gromov-Witten invariants on a moving elliptic curve, which now have been proven in collaboration with Pandharipande and Tseng, and are also connected to the failure of the Gorenstein conjecture on the moduli space of curves of compact type

The talk will be an overview of the intersection theory of $A_g$ and the moduli space of curves, and I will explain briefly the ideas behind some proofs.

This talk is part of the Algebraic Geometry Seminar series.

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