The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds

Abstract

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points. A nano-manifold is a Calabi-Yau threefold $X$ with very small Hodge numbers: $h^{1,1}(X)+h^{2,1}(X)\le 6$. We construct four rigid banana nano-manifolds ${\tilde{X}}_N$, $N\in \{5,6,8,9\}$, each with Hodge numbers given by $(h^{1,1},h^{2,1})=(4,0)$. We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus $g$ Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight $2g-2$ for a certain discrete subgroup $P_N^{\ast }\subset S{p}_4(\mathbb{R})$. We also compute the weight 4 modular form whose $p$th Fourier coefficient is given by the trace of the action of Frobenius on $H_{et}^3({\tilde{X}}_N,{\mathbb{Q}}_l)$ for almost all prime $p$. We observe that it is the unique weight 4 cusp form on $\hat{I}{“}_0(N)$.

Appendix with Mike Roth