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arXiv · 2206.01681

SYZ mirror symmetry for del Pezzo surfaces and affine structures

Siu-Cheong Lau·Tsung-Ju Lee·Yu-Shen Lin
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Abstract

We prove that the Landau--Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperK\"ahler 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 hyperK\"ahler rotation.

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Siu-Cheong Lau, Tsung-Ju Lee, Yu-Shen Lin. 2022-06-03. SYZ mirror symmetry for del Pezzo surfaces and affine structures. https://arxiv.org/abs/2206.01681

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