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

Abstract

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.