Stephen Pietromonaco, April 7, 2023

Title: Nano-Banana Manifolds and Siegel Paramodular Forms

Abstract: In this talk I will give new examples of Calabi-Yau threefolds (CY3s) which we call nano-banana manifolds. These are CY3s with small Hodge numbers---they are rigid, and they have only four independent curve classes. My main interest in these spaces comes from enumerative geometry, where one typically tries to compute curve-counting invariants, and investigates automorphic properties of the resulting generating series. This automorphy conjecturally sheds light on the mirror symmetry of your CY3. In the case of the nano-banana manifolds, I give a full computation of the Gromov-Witten potentials in three of the four curve classes, and show that these generating series transform like a Siegel paramodular form with level. From an arithmetic perspective, the nano-banana manifolds are defined over Q, and there is a weight 4 cusp modular form coming from counting points over finite fields. It remains an interesting open question what, if any, is the connection of this perspective to the enumerative geometry. This is joint work in progress with Jim Bryan.