Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 6

On the complex affine structures of SYZ-fibration of del Pezzo surfaces

Pages: 1837 – 1872

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n6.a7

Authors

Siu-Cheong Lau (Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, U.S.A.)

Tsung-Ju Lee (Center of Mathematical Sciences and Applications, Cambridge, Massachusetts, U.S.A.)

Yu-Shen Lin (Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, U.S.A.)

Abstract

Given any smooth cubic curve $E \subseteq \mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $\mathbb{P}^2 \: \backslash \: E$ constructed by Collins–Jacob–Lin [12] coincides with the affine structure used in Carl–Pomperla–Siebert [15] 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.

Published 30 June 2023