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Local mirror symmetry via SYZ

Benjamin Gammage

Abstract

In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.

Local mirror symmetry via SYZ

Abstract

In this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.

Paper Structure

This paper contains 4 sections, 14 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gluing microsheaf categories
  3. Calculating the skeleton
  4. SYZ geometry and cluster theory

Key Result

Theorem 1.2

There is an equivalence of categories $\textup{Fuk}(X_{m,n})\cong \textup{Coh}(X_{n,m}).$

Figures (1)

  • Figure 1: The plane $\mathcal{H},$ which lives inside the SYZ base of $X_{m,2}.$ The Lagrangian skeleton is the union of the torus fiber over the red point and a piece projecting to the blue shaded region. The higher-index critical points are depicted in green.

Theorems & Definitions (33)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 2.3
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.7: Nad-CnW
  • ...and 23 more