Markov numbers and Lagrangian cell complexes in the complex projective plane

TL;DR

This paper explores the relationship between Lagrangian embeddings of certain cell complexes called pinwheels in the complex projective plane and Markov numbers, revealing new constraints and classifications in symplectic geometry.

Contribution

It establishes a link between Lagrangian embeddings of pinwheels and Markov numbers, providing a complete characterization of when such embeddings exist and their disjointness properties.

Findings

Lagrangian pinwheels embed into CP^2 only if p is a Markov number.

A collection of disjoint Lagrangian pinwheels is limited to at most three, with p-values forming a Markov triple.

Results mirror a classification theorem for complex surfaces with quotient singularities.

Abstract

We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.