Markov numbers and Lagrangian cell complexes in the complex projective plane

Evans, Jonathan¹ ; Smith, Ivan²

¹ Department of Mathematics, University College London, London, United Kingdom
² Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom

Geometry & topology, Tome 22 (2018) no. 2, pp. 1143-1180.

Résumé

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 ${ℂ\mathbb{P}}^2$.

Détail

DOI : 10.2140/gt.2018.22.1143

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

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