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Family Floer SYZ singularities for the conifold transition

Hang Yuan

TL;DR

This work constructs a fully explicit non-archimedean SYZ bridge for the conifold, proving that the B-side (a crepant resolution) admits a tropically continuous fibration whose smooth part mirrors the A-side Lagrangian fibration on the smoothing, with singular fibers matching codimension-2 mirror data. It provides an explicit mirror Y over the Novikov field Λ and an algorithmic family Floer mirror X0^∨, glued from three affinoid charts with explicit wall-crossing, superpotential, and transition maps, and identifies an analytic embedding g that realigns the A- and B-side fibrations. The main result confirms, in the non-archimedean setting, the Chan–Pomerleano–Ueda speculation about the link between conifold smoothing and crepant resolution via SYZ, and it clarifies how codimension-2 missing points in mirror cluster varieties arise as dual SYZ singular fibers. The paper also highlights a precise Berkovich-geometric perspective via tropically continuous maps, and it raises broader questions about extending these constructions to cluster varieties and other higher-dimensional examples. Overall, the work delivers a concrete, highly explicit instance of SYZ duality with singular fibers in dimension three, offering new tools and viewpoints for non-archimedean mirror symmetry and cluster-structure phenomena.

Abstract

We show a mathematically precise version of the SYZ conjecture, proposed in the family Floer context, for the conifold with a conjectural mirror relation between smoothing and crepant resolution. The singular T-duality fibers are explicitly written and exactly correspond to the codimension-2 `missing points' in the mirror cluster variety, which confirms the speculation of Chan, Pomerleano, and Ueda but only in the non-archimedean setting. Concerning purely the area of Berkovich geometry and forgetting all the mirror symmetry background, our B-side analytic fibration is also a new explicit example of affinoid torus fibration with singular extension.

Family Floer SYZ singularities for the conifold transition

TL;DR

This work constructs a fully explicit non-archimedean SYZ bridge for the conifold, proving that the B-side (a crepant resolution) admits a tropically continuous fibration whose smooth part mirrors the A-side Lagrangian fibration on the smoothing, with singular fibers matching codimension-2 mirror data. It provides an explicit mirror Y over the Novikov field Λ and an algorithmic family Floer mirror X0^∨, glued from three affinoid charts with explicit wall-crossing, superpotential, and transition maps, and identifies an analytic embedding g that realigns the A- and B-side fibrations. The main result confirms, in the non-archimedean setting, the Chan–Pomerleano–Ueda speculation about the link between conifold smoothing and crepant resolution via SYZ, and it clarifies how codimension-2 missing points in mirror cluster varieties arise as dual SYZ singular fibers. The paper also highlights a precise Berkovich-geometric perspective via tropically continuous maps, and it raises broader questions about extending these constructions to cluster varieties and other higher-dimensional examples. Overall, the work delivers a concrete, highly explicit instance of SYZ duality with singular fibers in dimension three, offering new tools and viewpoints for non-archimedean mirror symmetry and cluster-structure phenomena.

Abstract

We show a mathematically precise version of the SYZ conjecture, proposed in the family Floer context, for the conifold with a conjectural mirror relation between smoothing and crepant resolution. The singular T-duality fibers are explicitly written and exactly correspond to the codimension-2 `missing points' in the mirror cluster variety, which confirms the speculation of Chan, Pomerleano, and Ueda but only in the non-archimedean setting. Concerning purely the area of Berkovich geometry and forgetting all the mirror symmetry background, our B-side analytic fibration is also a new explicit example of affinoid torus fibration with singular extension.
Paper Structure (25 sections, 5 theorems, 92 equations, 1 figure)

This paper contains 25 sections, 5 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Outline of the story
  4. Explicit mirror construction
  5. Additional evidence
  6. Lagrangian fibration on the deformed conifold: A side
  7. Lagrangian fibration
  8. Topological disks
  9. Action coordinates
  10. Topological embedding map $j$
  11. Family Floer T-duality construction
  12. General aspects of non-archimedean geometry
  13. Family Floer mirror construction: a quick review
  14. Void wall-crossing and explicit description
  15. Tropically continous fibration on the resolved conifold: B side
  16. ...and 10 more sections

Key Result

Theorem 1.1

$Y$ is SYZ mirror to $X$.

Figures (1)

  • Figure 1: The singular locus consists of two skew lines in $\mathbb R^3$

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 1.2
  • Conjecture 1.3
  • proof : Proof of Theorem \ref{['Main_theorem_SYZ_intro']} omitting (iii)
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • proof : Sketch of computation
  • Remark 2.1
  • ...and 13 more