Log Calabi-Yau mirror symmetry and non-archimedean disks
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910686939971584 |
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| author | Keel, Sean YU, Tony Yue |
| author_facet | Keel, Sean YU, Tony Yue |
| contents | We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of non-archimedean analytic disks. More generally, we studied the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log Calabi-Yau mirror symmetry and non-archimedean disks Keel, Sean YU, Tony Yue Algebraic Geometry Symplectic Geometry We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of non-archimedean analytic disks. More generally, we studied the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties. |
| title | Log Calabi-Yau mirror symmetry and non-archimedean disks |
| topic | Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2411.04067 |