Relative mirror symmetry for non-Fano varieties
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912648162967552 |
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| author | You, Fenglong |
| author_facet | You, Fenglong |
| contents | Given a smooth projective variety $X$ with a smooth anticanonical divisor $D$, we study mirror symmetry for the log Calabi--Yau pair $(X,D)$ without assuming that $D$ is nef. We consider the mirror proper Landau--Ginzburg model $(\check X,W)$ from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of $X$ and the classical period of $W$, and identify the discrepancy between them as originating from curve counts in $D$, governed by the mirror map associated with $D$. We also obtain an explicit formula for the proper potential $W$ that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative mirror symmetry for non-Fano varieties You, Fenglong Algebraic Geometry Given a smooth projective variety $X$ with a smooth anticanonical divisor $D$, we study mirror symmetry for the log Calabi--Yau pair $(X,D)$ without assuming that $D$ is nef. We consider the mirror proper Landau--Ginzburg model $(\check X,W)$ from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of $X$ and the classical period of $W$, and identify the discrepancy between them as originating from curve counts in $D$, governed by the mirror map associated with $D$. We also obtain an explicit formula for the proper potential $W$ that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential. |
| title | Relative mirror symmetry for non-Fano varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.13027 |