SYZ mirror symmetry for del Pezzo surfaces and affine structures

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.