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 1/p² (pq -- 1, 1) (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into CP² then p is a Markov number and we completely characterise q. We also show that a collection of Lagrangian pinwheels Lpᵢ;qᵢ , i = 1,...,N, cannot be made disjoint unless N ≤ 3 and the pᵢ 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 Q- Gorenstein smoothing whose general fibre is CP².