[Paper Review] On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Siu-Cheong Lau, Tsung-Ju Lee+1 more|arXiv (Cornell University)|May 11, 2020

Geometric and Algebraic Topology23 references4 citations

TL;DR

This paper establishes a direct correspondence between the complex affine structure of the SYZ fibration on the complement of a smooth cubic curve in $\mathbb{P}^2$ and the affine structure used in the Gross–Siebert program for mirror construction. Using Floer-theoretic gluing of immersed Lagrangians, the authors construct a mirror that agrees with the Carl–Pomperla–Siebert mirror, confirming a deep link between symplectic and algebraic mirror symmetry for 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.

Motivation & Objective

• To establish a correspondence between the complex affine structure of the SYZ fibration on $\mathbb{P}^2 \setminus E$ and the affine structure used in the Gross–Siebert mirror construction.
• To construct a mirror for $\mathbb{P}^2 \setminus E$ using immersed Lagrangians via Floer-theoretic gluing methods.
• To show that the mirror constructed via Floer theory agrees with the mirror constructed via the Gross–Siebert program.
• To demonstrate that the affine structure from the SYZ fibration matches the one arising from scattering diagrams in the algebraic mirror symmetry program.

Proposed method

• The authors use the special Lagrangian fibration on $\mathbb{P}^2 \setminus E$ constructed by Collins–Jacob–Lin to define a complex affine structure on the base.
• They identify this affine structure with the one used in Carl–Pomperla–Siebert's construction of the mirror via scattering diagrams.
• Using the family Floer homology framework, they construct a mirror via Maurer–Cartan elements of immersed Lagrangian tori.
• The gluing of local mirror charts is performed using Fukaya isomorphisms, incorporating quantum corrections from Maslov index zero holomorphic discs.
• The construction accounts for singular fibers, which source the Maslov index zero discs and drive the quantum corrections.
• The authors analyze the behavior of holomorphic discs and their boundary conditions to match the tropical disc counts with open Gromov–Witten invariants.

Experimental results

Research questions

• RQ1 Does the complex affine structure of the SYZ fibration on $\mathbb{P}^2 \setminus E$ coincide with the affine structure used in the Gross–Siebert mirror construction?
• RQ2 Can the mirror of $\mathbb{P}^2 \setminus E$ be constructed via Floer-theoretic gluing of immersed Lagrangians?
• RQ3 Is the mirror obtained via Floer theory equivalent to the mirror constructed via the Gross–Siebert program?
• RQ4 How do Maslov index zero holomorphic discs from singular fibers contribute to the quantum corrections in the mirror?
• RQ5 What is the role of the complex affine structure in relating symplectic and algebraic mirror symmetry for Del Pezzo surfaces?

Key findings

• The complex affine structure of the SYZ fibration on $\mathbb{P}^2 \setminus E$ matches exactly the affine structure used in the Carl–Pomperla–Siebert mirror construction.
• The mirror constructed via Floer-theoretic gluing of immersed Lagrangians agrees with the mirror from the Gross–Siebert program.
• The quantum corrections from Maslov index zero holomorphic discs are fully captured by the gluing process, matching the scattering diagram structure.
• The open Gromov–Witten invariants defined via the third author’s framework are identified with tropical disc counts, confirming consistency with the algebraic approach.
• The imaginary part of the mirror map $G(q)$ lies in $\sqrt{-1} \cdot \mathbb{R}_+$ for $q < 3$, and $\lim_{q \to -\infty} G(q) = \sqrt{-1} \cdot \infty$, confirming convergence and behavior at infinity.
• The analysis of holomorphic discs and their boundary behavior confirms that the locus of special Lagrangian fibers bounding such discs forms affine lines in the base, consistent with the affine structure.