Mirror Symmetry for log Calabi-Yau Surfaces II
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910368676184064 |
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| author | Lai, Jonathan Zhou, Yan |
| author_facet | Lai, Jonathan Zhou, Yan |
| contents | We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12703 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mirror Symmetry for log Calabi-Yau Surfaces II Lai, Jonathan Zhou, Yan Algebraic Geometry 14J32, 14J33, 14J10, 14J26 We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family. |
| title | Mirror Symmetry for log Calabi-Yau Surfaces II |
| topic | Algebraic Geometry 14J32, 14J33, 14J10, 14J26 |
| url | https://arxiv.org/abs/2201.12703 |