#### Volume 22, issue 2 (2018)

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Markov numbers and Lagrangian cell complexes in the complex projective plane

### Jonathan David Evans and Ivan Smith

Geometry & Topology 22 (2018) 1143–1180
##### 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 ${ℂℙ}^{2}$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels ${L}_{{p}_{i},{q}_{i}}$, $i=1,\dots ,N\phantom{\rule{0.3em}{0ex}}$, 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 $ℚ$–Gorenstein smoothing whose general fibre is ${ℂℙ}^{2}$.

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##### Keywords
symplectic four-manifolds and orbifolds, Markov numbers, Wahl singularities, vanishing cycles
##### Mathematical Subject Classification 2010
Primary: 14J17, 53D35, 53D42