On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Given any smooth cubic curve $E\subseteq {\mathbb{P}}^2$, we show that the complex affine structure of the special Lagrangian fibration of ${\mathbb{P}}^2\setminus E$ constructed by Collins–Jacob–Lin arXiv:1904.08363 coincides with the affine structure used in Carl–Pomperla–Siebert for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl–Pomperla–Siebert.