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

Technical Summary: Plumbings of Lens Spaces and Crepant Resolutions of Compound $A_n$ Singularities

Problem Statement
The paper investigates the symplectic topology of a specific class of 6-dimensional Weinstein manifolds constructed as cyclic plumbings of cotangent bundles of 3-dimensional lens spaces along circles. These manifolds, denoted $\mathcal{W}_f$, arise as the total spaces of Morse-Bott fibrations over $\mathbb{C}^*$ with singular fibers determined by a polynomial factorization $f(x,y) = \prod f_i(x,y)$. The central problem is to establish Homological Mirror Symmetry (HMS) for these non-compact Stein 3-folds and to understand their symplectic mapping class groups. Specifically, the authors aim to relate the derived wrapped Fukaya category of $\mathcal{W}_f$ to the derived category of coherent sheaves on the crepant resolution of a corresponding compound $A_n$ ($cA_n$) singularity $R_f = K[u,v,x,y]/(uv - f(x,y))$.

Methodology
The authors employ a combination of symplectic geometry, algebraic geometry, and category theory:

1. Symplectic Construction and Computation:

   • The manifold $\mathcal{W}_f$ is realized as an affine $A_n$ plumbing of cotangent bundles $T^*Q_i$ (where $Q_i$ are lens spaces or $S^1 \times S^2$) along circles.
   • The wrapped Fukaya category $\mathcal{W}(\mathcal{W}_f; K)$ is computed by counting holomorphic sections of the Morse-Bott fibration $\mathcal{W}_f \to \mathbb{C}^*$. The authors utilize a specific wrapping Hamiltonian adapted to the fibration structure.
   • They identify the endomorphism $A_\infty$-algebra of a generating set of Lagrangian fibers ($L_0, \dots, L_n$) with a path algebra of a quiver with relations, deformed by the polynomial $f(x,y)$.

2. Algebraic Geometry and Mirror Symmetry:

   • On the B-side, the authors construct a specific crepant resolution $\phi: Y_f \to \text{Spec}(R_f)$ of the $cA_n$ singularity. The exceptional locus consists of a chain of rational curves.
   • They utilize the Evans-Lekili model, which relates relative wrapped Fukaya categories of punctured discs to path algebras of quivers.
   • The authors perform categorical localizations and completions (with respect to word-length filtrations and Seidel elements) to bridge the gap between the non-compact affine setting and the complete local setting.

3. Mapping Class Groups:

   • The authors analyze the symplectic mapping class group $\pi_0 \text{Symp}_c(\mathcal{W}_f)$ by leveraging the HMS equivalence. They relate autoequivalences of the derived category of coherent sheaves on the exceptional locus (computed by Toda) to symplectic automorphisms of $\mathcal{W}_f$.

Key Contributions and Results

1. Homological Mirror Symmetry (HMS) Theorems:
   The paper proves two versions of HMS for the Stein 3-fold $\mathcal{W}_f$:

   • The Uncompleted Version (Theorem 3): There is an equivalence between the derived wrapped Fukaya category of $\mathcal{W}_f$ and the bounded derived category of coherent sheaves on the complement of a divisor in the crepant resolution:
     $$D^b\text{Coh}(\mathcal{Y}_f) \simeq D^{\text{perf}}\mathcal{W}(\mathcal{W}_f; K)$$
     where $\mathcal{Y}_f = Y_f \setminus D$.
   • The Completed Version (Theorem 7): There is an equivalence between the derived category of perfect modules over the completed wrapped Fukaya category (completed with respect to the word-length filtration of Hamiltonian chords) and the bounded derived category of coherent sheaves on the complete local version of the resolution $\hat{Y}_f$:
     $$D^{\text{perf}}\widehat{\mathcal{W}}(\mathcal{W}_f; K) \simeq D^b\text{Coh}(\hat{Y}_f)$$

2. Relative Singularity Categories and Multi-Bubble Plumbings:
   By filling in the origin of the base of the fibration, the authors obtain a "multi-bubble" plumbing $W_f$. They prove (Theorem 10) that the derived wrapped Fukaya category of $W_f$ is equivalent to the relative singularity category of the crepant resolution $Y_f$ (defined as the quotient of the derived category by the structure sheaf), provided a specific non-isotopy condition on the Lagrangian cores holds. This generalizes previous results on double bubble plumbings.

3. Symplectic Mapping Class Groups:
   The paper establishes that the symplectic mapping class group of $\mathcal{W}_f$ is infinitely generated.

   • Theorem 15: If the $cA_n$ singularity is isolated, a specific infinitely generated subgroup of the pure braid group $PBr_{n+2}$ (denoted $PBr^c_{n+2}$) split-injects into the compactly supported symplectic mapping class group $\pi_0 \text{Symp}_c(\mathcal{W}_f)$.
   • Theorem 17: A similar split injection exists for the graded symplectic mapping class group involving the full pure braid group $PBr_{n+2}$.
   • These results generalize the work of Keating-Smith (who treated the conifold case) to the broader class of lens space plumbings.

4. Applications to Lagrangian Embeddings:
   Using the existence of cyclic quasi-dilations derived from the HMS equivalence, the authors prove (Proposition 47) that under certain conditions, no closed oriented exact Lagrangian submanifold in $W_f$ can be a $K(\pi, 1)$ space.

Significance and Claims
The paper claims to generalize the homological mirror symmetry results for double bubble plumbings (Smith-Wemyss) and conifold smoothings (Keating-Smith) to a cyclic class of plumbings involving lens spaces.

• It provides a concrete realization of HMS for a class of 3-folds where the mirror geometry involves non-trivial lens space cores and cyclic intersection patterns.
• It partially answers the "realization question" posed by Smith and Wemyss regarding which relative singularity categories arise from plumbings of lens spaces.
• It demonstrates that the symplectic mapping class groups of these Stein manifolds are rich and infinitely generated, extending the known phenomenon from the $A_1$ (conifold) case to higher $A_n$ and more complex topological types.
• The work confirms a conjecture by Lekili-Segal regarding the relation between the wrapped Fukaya category of a total space of an algebraic torus fibration and that of its base in this specific setting.

The authors emphasize that their results rely on the computation of the wrapped Fukaya category via Morse-Bott fibrations and the subsequent identification of the resulting algebra with the path algebra of a quiver associated with the crepant resolution. The proofs of the mapping class group results depend on Toda's computation of the autoequivalence group of the derived category of coherent sheaves on the exceptional locus.