SYZ mirror symmetry for del Pezzo surfaces and affine structures

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 P¹×P¹ [16] and compare it with the integral affine structures used in the work of Carl–Pumperla–Siebert [9]. 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 [13] and Hong–Kim–Lau [41]. They agree with the result of limit of hyperKähler rotations.