Markov numbers and Lagrangian cell complexes in the complex projective plane

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 $\left(1∕p^2\right)\left(pq-1,1\right)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into ${ℂ\mathbb{P}}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\dots ,N$, cannot be made disjoint unless $N\le 3$ 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 $\mathbb{Q}$–Gorenstein smoothing whose general fibre is ${ℂ\mathbb{P}}^2$.