2024/05/07 by Jim Bryan, Bryan, Jim, Stephen Pietromonaco +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2405.04701
openalex publication_date 2024/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: h1,1(X)+h2,1(X)≤ 6. We construct four rigid banana nano-manifolds XN, N∈ \5,6,8,9 \, each with Hodge numbers given by (h1,1,h2,1)=(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 ⊂ Sp4(ℝ). We also compute the weight 4 modular form whose pth Fourier coefficient is given by the trace of the action of Frobenius on H3et (XN ,ℚl) for almost all prime p. We observe that it is the unique weight 4 cusp form on Γ0(N).