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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

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 (1p2)(pq 1,1) (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into 2 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 –Gorenstein smoothing whose general fibre is 2.

symplectic four-manifolds and orbifolds, Markov numbers, Wahl singularities, vanishing cycles
Mathematical Subject Classification 2010
Primary: 14J17, 53D35, 53D42
Received: 12 July 2016
Revised: 3 May 2017
Accepted: 11 June 2017
Published: 16 January 2018
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, András I Stipsicz
Jonathan David Evans
Department of Mathematics
University College London
United Kingdom
Ivan Smith
Centre for Mathematical Sciences
University of Cambridge
United Kingdom