Séminaire des jeunes : Aitor Iribar López (Université de Zurich) : Intersection theory on moduli spaces of abelian varieties
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Séminaire des jeunes : Aitor Iribar López (Université de Zurich) : Intersection theory on moduli spaces of abelian varieties
25 mars / 16:30 - 17:30
Résumé :
We study A_g, the moduli space of principally polarized abelian varieties of dimension g. The subring generated by the Chern classes of the Hodge bundle is called the tautological ring, and it 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 × A_5] is not tautological in A_6, and later I showed that [A_1 × A_{g-1}] is not tautological for g=12 or g ≥ 16 even.
The cycle [A_1 × 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 λg conjecture on the moduli space of curves.
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.
Séminaire des jeunes : Aitor Iribar López (Université de Zurich) : Intersection theory on moduli spaces of abelian varieties