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Plumbings of lens spaces and crepant resolutions of compound $A_n$ singularities

Bilun Xie, Yin Li

TL;DR

<3-5 sentence high-level summary>

Abstract

We prove that the completed derived wrapped Fukaya categories of certain affine $A_n$ plumbings $W_f^\circ$ of $3$-dimensional lens spaces along circles are equivalent to the derived categories of coherent sheaves on crepant resolutions of the corresponding compound $A_n$ ($cA_n$) singularities $\mathbb{C}[\![u,v,x,y]\!]/(uv-f(x,y))$. The proof relies on the verification of a conjecture of Lekili-Segal. After localization, we obtain an equivalence between the derived wrapped Fukaya category of the (non-affine) $A_n$ plumbing $W_f\supset W_f^\circ$ of lens spaces along circles and the relative singularity category of the crepant resolution. This generalizes the result of Smith-Wemyss and partially answers their realization question. As a consequence, we obtain a faithful representation of the pure braid group $\mathit{PBr}_{n+1}$ on the graded symplectic mapping class group of $W_f$ when the corresponding $cA_n$ singularity is isolated.

Plumbings of lens spaces and crepant resolutions of compound $A_n$ singularities

TL;DR

<3-5 sentence high-level summary>

Abstract

We prove that the completed derived wrapped Fukaya categories of certain affine plumbings of -dimensional lens spaces along circles are equivalent to the derived categories of coherent sheaves on crepant resolutions of the corresponding compound () singularities . The proof relies on the verification of a conjecture of Lekili-Segal. After localization, we obtain an equivalence between the derived wrapped Fukaya category of the (non-affine) plumbing of lens spaces along circles and the relative singularity category of the crepant resolution. This generalizes the result of Smith-Wemyss and partially answers their realization question. As a consequence, we obtain a faithful representation of the pure braid group on the graded symplectic mapping class group of when the corresponding singularity is isolated.

Paper Structure

This paper contains 20 sections, 29 theorems, 131 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Plumbings of cotangent bundles
  4. Crepant resolutions of $cA_n$ singularities
  5. Seidel representation
  6. Legendrian surgery
  7. Computing the wrapped Fukaya category
  8. The setup
  9. The case of a single Lagrangian
  10. Convention.
  11. Products between two adjacent Lagrangians
  12. Concluding the computation
  13. Homological mirror symmetry and autoequivalences
  14. The Evans-Lekili model
  15. Proofs of the results
  16. ...and 5 more sections

Key Result

Theorem 3

Let $\mathbb{K}$ be any field, and suppose that the pairs $(k_i,\pm l_i)\in\mathbb{Z}_{\geq0}\times\mathbb{Z}$ for $i=0,\cdots,n$ are as in (eq:kl). Then there is an equivalence where $D^\mathit{perf}\widehat{\mathcal{W}}(\ring{W}_f;\mathbb{K})$ is the derived category of perfect $A_\infty$-modules over $\widehat{\mathcal{W}}(\ring{W}_f;\mathbb{K})$ and $D^b\mathit{Coh}(\widehat{Y}_f)$ is the bou

Figures (12)

  • Figure 1: Base of the Morse-Bott fibration $\ring{\pi}$ when $n=5$. The arcs $\alpha_i$ (in teal) connecting the critical values $c_i$ and $c_{i+1\textrm{ mod }n}$ are matching paths of the Lagrangians $Q_0,\cdots,Q_n$, while the arcs $\gamma_0,\cdots,\gamma_5$ (in orange) split-generate the relative wrapped Fukaya category $\mathcal{W}(T^\ast S^1,D_5)$.
  • Figure 2: Gradient vector field and flow lines (in orange) of the function (\ref{['eq:potential']}) when $n=3$ (picture created by ChatGPT)
  • Figure 3: Wrapping in the cylinder
  • Figure 4: The triangle product of generators above $x_0$ and $x_1$, where the upper edge and the lower edge of the strip are identified. The shaded triangle corresponds precisely to the one in Figure \ref{['fig:wrapping']}.
  • Figure 5: The triangle product of generators above $x_{-1}$ and $x_1$
  • ...and 7 more figures

Theorems & Definitions (60)

  • Example 1
  • Example 2
  • Theorem 3
  • Remark 4
  • Proposition 5
  • Theorem 6
  • Remark 7
  • Proposition 8
  • Corollary 9
  • Remark 10
  • ...and 50 more