Niklas Ludwig: Modularity of Refined Humbert Loci in A_3

Abstract:

We consider refined Humbert loci in the moduli space of principally polarized abelian threefolds \(\mathcal{A}_3\), where the abelian variety contains an extra class in the Neron–Severi group with prescribed characteristic polynomial and prescribed type. Recently, Oberdieck formulated a conjecture about the quasi-modularity of the generating series of refined Humbert loci weighted by reduced Gromov–Witten invariants. We explain this conjecture in the case of \(\mathcal{A}_3\) and discuss some approaches towards its proof.