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Markov numbers and Lagrangian cell complexes in the complex projective plane

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>16/01/2018
<mark>Journal</mark>Geometry and Topology
Issue number2
Volume22
Number of pages37
Pages (from-to)1143-1180
Publication statusPublished
Original languageEnglish

Abstract

We study Lagrangian embeddings of a class of two-dimensional cell complexes Lp,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.