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 $\frac{1}{p^2}(pq-1,1)$ (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 $L_{p_i,q_i}$, $i=1,\dots ,N$, 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 $\mathbf{Q}$-Gorenstein smoothing whose general fibre is ${\mathbf{CP}}^2$.

32 pages, 3 figures; v2 corrected some typos and added clarifications; v3 incorporated further corrections and comments. To appear in Geometry and Topology

References in corpus (2)

• Degenerations of del Pezzo surfaces I
• Symplectic Rational Blow-up

Cited by in corpus (6)

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• Horizontal decompositions, I
• On Stein rational balls smoothly but not symplectically embedded in CP^2
• Weinstein neighbourhood theorems for stratified subspaces
• Horizontal decompositions, II
• Simple embeddings of rational homology balls and antiflips