The nearby Lagrangian conjecture for pinwheels

Technical Summary: The Nearby Lagrangian Conjecture for Pinwheels

Problem Statement
The paper addresses the Nearby Lagrangian Conjecture (NLC), attributed to V. I. Arnol'd, which posits that any exact Lagrangian in the cotangent bundle $T^*Q$ of a smooth closed manifold $Q$ is Hamiltonian isotopic to the zero section. While the conjecture has been established for several low-dimensional cases (e.g., $T^*S^2$, $T^*T^2$) using pseudoholomorphic curve techniques, it remains open for general manifolds and, crucially, for singular Lagrangians.

The authors focus on a specific class of singular Lagrangians called Lagrangian $(p, q)$-pinwheels. A $(p, q)$-pinwheel is a topological $p$-pinwheel (a 2-cell attached to a 1-cell via a $p$-to-1 covering) where the 2-cell is Lagrangian and meets the core circle in a specific geometric manner determined by $q$. These objects naturally arise as the Lagrangian skeletons ($L^{\text{st}}_{p,q}$) of rational homology balls $B_{p,q}$, which are the Milnor fibers of cyclic quotient singularities $\frac{1}{p^2}(1, pq-1)$.

The central problem is to determine whether any Lagrangian $(p, q)$-pinwheel embedded in $B_{p,q}$ is Hamiltonian isotopic to the standard skeleton $L^{\text{st}}_{p,q}$. This is the "NLC1" subconjecture adapted to the singular setting of $B_{p,q}$.

Methodology
The proof is structured into two largely independent technical parts, combining symplectic topology, algebraic geometry, and pseudoholomorphic curve theory.

1. Symplectic Uniqueness (Part 1):

   • Compactification: The authors construct a compactification $X_{p,q}$ of the non-compact manifold $B_{p,q}$. This is achieved by capping the almost toric base diagram of $B_{p,q}$ and resolving the resulting orbifold singularity. The resulting space $X_{p,q}$ is birationally derived from a Hirzebruch surface.
   • Regulations and Neck-Stretching: They utilize the concept of "regulations" (foliations by rational curves) on $X_{p,q}$. By performing a symplectic rational blow-up along an embedded pinwheel, they analyze the behavior of pseudoholomorphic curves under neck-stretching.
   • Persistence of Foliations: A key technical result is showing that the regulation (a foliation by spheres of square zero) on $X_{p,q}$ "persists" after the rational blow-up. Specifically, the blown-up manifold $\tilde{X}$ admits a unique broken ruling consisting of the Wahl chain (exceptional spheres from the blow-up) and the compactifying divisor, connected by a single exceptional sphere.
   • Symplectomorphism Construction: Using the uniqueness of these regulations and the fact that the complements of the pinwheel neighborhoods are minimal, they construct a compactly supported symplectomorphism $\psi$ mapping any Lagrangian pinwheel $L$ to the standard skeleton $L^{\text{st}}_{p,q}$.

2. Hamiltonian Isotopy (Part 2):

   • Symplectomorphism Groups: To upgrade the symplectomorphism $\psi$ to a Hamiltonian isotopy, the authors compute the weak homotopy type of the group of compactly supported symplectomorphisms, $\text{Symp}_c(B_{p,q})$.
   • Comb Diagrams: They employ "Comb diagrams," a tool relating the symplectomorphism group of a manifold to that of a submanifold and its complement. They relate $\text{Symp}_c(B_{p,q})$ to the group $\text{Symp}_c(W_{p,q})$, where $W_{p,q} = B_{p,q} \setminus K_{p,q}$ (the complement of the central cylinder).
   • Liouville Completion: They identify $W_{p,q}$ as a star-shaped domain in $T^*Wh$, the cotangent bundle of the Whitney sphere. Using results on the contractibility of symplectomorphism groups of such domains, they show that $\text{Symp}_c(W_{p,q})$ is weakly contractible.
   • The Pintwist: They demonstrate that $\text{Symp}_c(B_{p,q})$ is weakly homotopy equivalent to $\mathbb{Z}$, generated by a specific symplectomorphism called the pintwist $\tau_{p,q}$. This map is the symplectic monodromy associated with the smoothing of the underlying singularity.
   • Conclusion: Since the pintwist preserves the Lagrangian skeleton up to Hamiltonian isotopy (as shown via Banyaga's extension theorem and vanishing cohomology), any symplectomorphism mapping $L$ to $L^{\text{st}}_{p,q}$ can be composed with a power of the pintwist to yield a Hamiltonian isotopy.

Key Contributions and Results

• Main Theorem (Theorem A): Any Lagrangian $(p, q)$-pinwheel in $B_{p,q}$ is Hamiltonian isotopic to the standard Lagrangian skeleton $L^{\text{st}}_{p,q}$. This establishes the Nearby Lagrangian Conjecture for this wide class of singular Lagrangians.
• Symplectic Classification (Theorem B): Any two Lagrangian $(p, q)$-pinwheels in $B_{p,q}$ are related by a compactly supported symplectomorphism.
• Symplectomorphism Group Structure (Theorem C): The group $\text{Symp}_c(B_{p,q})$ is weakly homotopy equivalent to $\mathbb{Z}$, generated by the pintwist $\tau_{p,q}$. This provides the first computation of this group for $p > 2$ (previous results existed only for $p=1, 2$).
• Applications:
  1. Pin-Ball Non-Squeezing: The authors prove a Gromov non-squeezing theorem for "pin-balls" $B_{p,q}(r)$, showing they embed into pin-cylinders $E_{p,q}(\alpha, \infty)$ if and only if $r \le \alpha$. The obstruction arises from the area of exceptional curves in the rational blow-up.
  2. Local Unknotting: They provide a new proof of the Eliashberg–Polterovich theorem on the local unknotting of Lagrangian planes in $\mathbb{R}^4$, viewing the $(1,1)$-pinwheel (a Lagrangian disc) as a special case.
  3. Uniqueness of Pinwheel Types: They show that the only Lagrangian pinwheels embeddable in $B_{p,q}$ are of type $(p, q)$ (or $(p, p-q)$), extending previous results on the non-existence of smooth exact Lagrangians in these balls.

Significance and Claims
The paper claims to resolve the Nearby Lagrangian Conjecture for a significant class of singular Lagrangians that serve as the "zero sections" for rational homology balls. The authors emphasize that their approach leverages the interplay between symplectic geometry and the algebraic geometry of surface singularities (specifically Wahl singularities).

The significance lies in:

• Extending NLC: Moving beyond smooth manifolds to singular Lagrangians, which are natural objects in the study of symplectic singularities and mirror symmetry.
• New Techniques: The successful application of "regulations" and neck-stretching in the context of rational homology balls, bypassing the need for a global compactification of the ambient space in the same way as previous works on $T^*Q$.
• Group Computations: Providing a complete description of the symplectic mapping class group of $B_{p,q}$, revealing a structure generated by a natural geometric operation (the pintwist) analogous to Dehn twists.
• Quantitative Obstructions: Demonstrating that the methods used for uniqueness also yield quantitative embedding obstructions (non-squeezing), linking the topology of the pinwheels to symplectic capacities.

The authors note that their work serves as a "prequel" to broader embedding theorems for rational homology balls into $\mathbb{CP}^2$, providing the local uniqueness results necessary for those global classifications. They explicitly state that their methods rely on the specific geometry of $B_{p,q}$ and the almost toric fibrations, and do not immediately generalize to all dimensions or all singular Lagrangians without further development.