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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.04701 |
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| _version_ | 1866929337607913472 |
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| author | Bryan, Jim Pietromonaco, Stephen |
| author_facet | Bryan, Jim Pietromonaco, Stephen |
| contents | 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 $Γ_{0}(N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds Bryan, Jim Pietromonaco, Stephen Algebraic Geometry 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 $Γ_{0}(N)$. |
| title | The Enumerative Geometry and Arithmetic of Banana Nano-Manifolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2405.04701 |