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Floer theory for the variation operator of an isolated singularity

Hanwool Bae, Cheol-Hyun Cho, Dongwook Choa, Wonbo Jeong

TL;DR

The paper develops a symplectic framework for singularity theory by constructing a Floer-theoretic variation operator $\mathcal{V}$ and a monodromy Lagrangian Floer cohomology $HF^*_{\rho}$ that categorify classical Seifert-type data. Central to the construction is the monodromy class $\Gamma \in SH^*(\rho^{-1})$ and the twisted closed–open map, which together produce a cone formalism that yields a Floer-theoretic variation on Lagrangians in the Milnor fiber. The authors prove a range of results linking $HF^*_{\rho}$ to Seifert pairing, establish invariance under Hamiltonian data, and connect to the Fukaya category via an adapted family and exceptional collections, with explicit depth-zero plane-curve examples including $E_6$ and $A_n$. Overall, the work provides a robust categorified bridge between singularity theory and symplectic topology, enabling new insights into monodromy, variation, and mirror-symmetry-type correspondences.

Abstract

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class $Γ$ in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

Floer theory for the variation operator of an isolated singularity

TL;DR

The paper develops a symplectic framework for singularity theory by constructing a Floer-theoretic variation operator and a monodromy Lagrangian Floer cohomology that categorify classical Seifert-type data. Central to the construction is the monodromy class and the twisted closed–open map, which together produce a cone formalism that yields a Floer-theoretic variation on Lagrangians in the Milnor fiber. The authors prove a range of results linking to Seifert pairing, establish invariance under Hamiltonian data, and connect to the Fukaya category via an adapted family and exceptional collections, with explicit depth-zero plane-curve examples including and . Overall, the work provides a robust categorified bridge between singularity theory and symplectic topology, enabling new insights into monodromy, variation, and mirror-symmetry-type correspondences.

Abstract

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.
Paper Structure (45 sections, 55 theorems, 249 equations, 25 figures)

This paper contains 45 sections, 55 theorems, 249 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Relation to pertinent works
  3. Acknowledgements
  4. Symplectic cohomology of an exact symplectomorphism
  5. Fixed point Floer cohomology
  6. Fixed point Floer cohomologies for linear Hamiltonians
  7. Perturbation with Reeb flows
  8. Symplectic cohomology of $\phi$
  9. Monodromy class $\Gamma$ of an isolated singularity
  10. Milnor fiber and Monodromy $\rho$ of an isolated singularity
  11. Floer complex for $\rho^{-1}$ and its Hamiltonian perturbation $\check{\rho}^{-1}$
  12. Monodromy class $\Gamma$
  13. Comparison with the spectral sequence of McLeanMc19
  14. Grading
  15. Variation operator $\mathcal{V}$ via the closed-open map
  16. ...and 30 more sections

Key Result

Theorem 1.3

(See Corollary cor:VLisproper) For every object $L$ of $\mathcal{WF}(M)$, $\mathcal{V}(L)$ is a proper object, i.e., for any $L'$, both $HW^*\left(\mathcal{V}(L), L'\right)$ and $HW^*\left(L', \mathcal{V}(L)\right)$ are finite-dimensional.

Figures (25)

  • Figure 1:
  • Figure 2: Examples of divide and $A\Gamma$ diagram
  • Figure 3: Vanishing cycles and adapted family of $E_{6}$
  • Figure 4: Illustration of the image of $\rho$ and $\check\rho$ for the curve $L$ in the case $n=2$.
  • Figure 5: Illustration of the Morse-Bott family $\Sigma$ for $n=2$ case
  • ...and 20 more figures

Theorems & Definitions (122)

  • Definition 1.1: see Definition \ref{['defn:gamma']}
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6: See Theorem \ref{['thm: HF is finite']}
  • Theorem 1.7: Arnold, Gussein-Zade, Varchenko AGV2
  • Theorem 1.8: See Theorem \ref{['prop:HF and V']} and \ref{['prop:Serre dual']}
  • Remark 1.9
  • Definition 1.10
  • ...and 112 more