香蕉纳米流形的枚举几何与算术
The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds
AI总结：
本文构造四个 Hodge 数为 (4,0) 的刚性香蕉纳米 Calabi–Yau 三维流形，并计算其 Donaldson–Thomas、Gromov–Witten 及 Frobenius 迹相关模形式。
摘要：
香蕉流形是一个以 Abel 曲面为纤维的 Calabi–Yau 三维流形，其奇异纤维包含香蕉构型：三条有理曲线在两个点彼此相交。纳米流形是指具有极小 Hodge 数的 Calabi–Yau 三维流形 X，满足 h^{1,1}(X)+h^{2,1}(X)≤6。我们构造了四个刚性香蕉纳米流形 \ ilde{X}_N，其中 N∈{5,6,8,9}，每个的 Hodge 数均为 (h^{1,1},h^{2,1})=(4,0)。我们计算了香蕉曲线类的 Donaldson–Thomas 配分函数，并证明相关的亏格 g Gromov–Witten 势是关于某个离散子群 P^{*}_{N}⊂Sp_4(R)、权为 2g−2 的亏格 2 亚纯 Siegel 模形式。我们还计算了权 4 模形式，其第 p 个 Fourier 系数由 Frobenius 在 H^3_et(\ ilde{X}_N,Q_l) 上作用的迹给出，适用于几乎所有素数 p。我们观察到，它是 Γ_0(N) 上唯一的权 4 尖点形式。
英文摘要：
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: $h^{1,1}(X)+h^{2,1}(X)\leq 6$. 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 $P^{*}_{N} \subset Sp_{4}(\mathbb{R})$. We also compute the weight 4 modular form whose $p$th Fourier coefficient is given by the trace of the action of Frobenius on $H^{3}_{et }(\tilde{X}_N ,{\mathbb{Q}}_{l})$ for almost all prime $p$. We observe that it is the unique weight 4 cusp form on $Γ_{0}(N)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
