Family Floer mirror space for local SYZ singularities

TL;DR

This paper rigorously proves the SYZ conjecture for toric Calabi-Yau manifolds with singular fibers, explicitly constructs the dual fibration, and explores new insights into SYZ singularities and mirror symmetry.

Contribution

It provides the first precise mathematical realization of the SYZ conjecture with singular fibers beyond topology, including explicit dual fibrations and new perspectives on SYZ singularities.

Findings

Explicit dual singular fibration constructed and verified.

Maurer-Cartan set of singular Lagrangian is a strict subset of dual fiber.

Supports conjectures in Landau-Ginzburg mirror models.

Abstract

We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with singular fibers beyond the topological level. The dual singular fibration is explicitly written and proved to be compatible with the family Floer mirror construction. Moreover, we discover that the Maurer-Cartan set of a singular Lagrangian is only a strict subset of the corresponding dual singular fiber. This responds negatively to the previous expectation and leads to new perspectives of SYZ singularities. As extra evidence, we also check some computations for a well-known folklore conjecture for the Landau-Ginzburg model.