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Mirror symmetry and Fukaya categories of singular hypersurfaces

Maxim Jeffs

TL;DR

This work provides a principled A-model construction for the Fukaya category of singular hypersurfaces by localizing the Fukaya category of nearby fibers at Seidel's natural transformation, and proves an A-model analog of Orlov's Knörrer periodicity: for a smooth affine $X$ and a polynomial $f$ with a single critical fiber, $D^{\pi}\mathcal{W}(f^{-1}(0))$ is equivalent to $D^{\pi}\mathcal{W}(X\times\mathbb{C},zf)$ after inverting $s$. The authors establish an upgraded Abouzaid–Auroux–Katzarkov equivalence between $\mathcal{W}(f^{-1}(t))$ and $\mathcal{W}(X\times\mathbb{C},z(f-t))$, extend it to relative/fiberwise-stop contexts, and then deduce the Knörrer-type equivalence via stop-removal and generation, yielding HMS-type statements at large complex structure limits for abelian varieties. The paper also demonstrates concrete mirror-symmetry consequences, including the Tower of Pants, nodal curves, and elliptic curves, and discusses generalizations in the Gross–Siebert program linking large complex/volume limits through the family Floer framework. Overall, it provides a robust bridge between singular symplectic geometry, LG models, and mirror symmetry, with potential extensions to complete intersections and logarithmic structures.

Abstract

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, we prove an A-side analog of Orlov's derived Knörrer periodicity theorem by showing that Auroux's category is derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, we describe how this definition implies homological mirror symmetry for some large complex structure limit degenerations of abelian varieties.

Mirror symmetry and Fukaya categories of singular hypersurfaces

TL;DR

This work provides a principled A-model construction for the Fukaya category of singular hypersurfaces by localizing the Fukaya category of nearby fibers at Seidel's natural transformation, and proves an A-model analog of Orlov's Knörrer periodicity: for a smooth affine and a polynomial with a single critical fiber, is equivalent to after inverting . The authors establish an upgraded Abouzaid–Auroux–Katzarkov equivalence between and , extend it to relative/fiberwise-stop contexts, and then deduce the Knörrer-type equivalence via stop-removal and generation, yielding HMS-type statements at large complex structure limits for abelian varieties. The paper also demonstrates concrete mirror-symmetry consequences, including the Tower of Pants, nodal curves, and elliptic curves, and discusses generalizations in the Gross–Siebert program linking large complex/volume limits through the family Floer framework. Overall, it provides a robust bridge between singular symplectic geometry, LG models, and mirror symmetry, with potential extensions to complete intersections and logarithmic structures.

Abstract

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, we prove an A-side analog of Orlov's derived Knörrer periodicity theorem by showing that Auroux's category is derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, we describe how this definition implies homological mirror symmetry for some large complex structure limit degenerations of abelian varieties.

Paper Structure

This paper contains 14 sections, 25 theorems, 76 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Knörrer Periodicity
  3. Mirror Symmetry
  4. Outline of Proof of Main Theorem
  5. Definitions and Conventions
  6. Cap and Cup Functors
  7. Complements of Fibers
  8. Knörrer Periodicity for Hypersurfaces
  9. The AAK Equivalence
  10. Proof of Main Theorem
  11. Applications to Mirror Symmetry
  12. The Tower of Pants
  13. Applications to Curves
  14. Generalizations

Key Result

THEOREM 1

(Derived Knörrer Periodicity) Suppose that $X$ is a smooth affine variety with an embedding $X \to \mathbb{C}^N$ inducing a Stein structure on $X$, and suppose $f:X \to \mathbb{C}$ is the restriction of a polynomial function on $\mathbb{C}^N$. If $f$ has a single critical fiber $f^{-1}(0)$ then for

Figures (11)

  • Figure 1: The cup functor.
  • Figure 2: A Lagrangian cobordism $L$ inside a concatenation of copies of $X$.
  • Figure 3: Skeleton in Example \ref{['example']} (in blue).
  • Figure 4: The handle attachment procedure from Proposition \ref{['prop:surgery']}: cores of additional handles are in red and cocores in blue. Length of vectors not to scale.
  • Figure 5: Producing linking disks of $F$ from objects $L$ (in red) and linking disks of $\Lambda$ (in green).
  • ...and 6 more figures

Theorems & Definitions (52)

  • DEFINITION 1
  • THEOREM 1
  • DEFINITION 2
  • CONJECTURE 1
  • THEOREM 2
  • THEOREM 3
  • Remark 1
  • THEOREM 4
  • THEOREM 5
  • CONJECTURE 2
  • ...and 42 more