SYZ mirror symmetry for del Pezzo surfaces and affine structures
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917629083516928 |
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| author | Lau, Siu-Cheong Lee, Tsung-Ju Lin, Yu-Shen |
| author_facet | Lau, Siu-Cheong Lee, Tsung-Ju Lin, Yu-Shen |
| contents | We prove that the Landau--Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperKähler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine structure of the special Lagrangian fibrations constructed by Collins--Jacob--Lin in $\mathbf{P}^1\times \mathbf{P}^1$ arXiv:1904.08363 and compare it with the integral affine structures used in the work of Carl--Pumperla--Siebert arXiv:2205.07753. We also construct the Floer-theoretical Landau--Ginzburg mirrors of smoothing of $A_n$-singularities and monotone del Pezzo surfaces, by using the gluing method of Cho--Hong--Lau arXiv:1810.02045 and Hong--Kim--Lau arXiv:1805.11738. They agree with the result of hyperKähler rotation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01681 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | SYZ mirror symmetry for del Pezzo surfaces and affine structures Lau, Siu-Cheong Lee, Tsung-Ju Lin, Yu-Shen Algebraic Geometry Differential Geometry Symplectic Geometry 32Q25, 53D37 We prove that the Landau--Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperKähler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine structure of the special Lagrangian fibrations constructed by Collins--Jacob--Lin in $\mathbf{P}^1\times \mathbf{P}^1$ arXiv:1904.08363 and compare it with the integral affine structures used in the work of Carl--Pumperla--Siebert arXiv:2205.07753. We also construct the Floer-theoretical Landau--Ginzburg mirrors of smoothing of $A_n$-singularities and monotone del Pezzo surfaces, by using the gluing method of Cho--Hong--Lau arXiv:1810.02045 and Hong--Kim--Lau arXiv:1805.11738. They agree with the result of hyperKähler rotation. |
| title | SYZ mirror symmetry for del Pezzo surfaces and affine structures |
| topic | Algebraic Geometry Differential Geometry Symplectic Geometry 32Q25, 53D37 |
| url | https://arxiv.org/abs/2206.01681 |