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2018 Markov numbers and Lagrangian cell complexes in the complex projective plane
Jonathan David Evans, Ivan Smith
Geom. Topol. 22(2): 1143-1180 (2018). DOI: 10.2140/gt.2018.22.1143

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 (1p2)(pq1,1) (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 N3 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.

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Jonathan David Evans. Ivan Smith. "Markov numbers and Lagrangian cell complexes in the complex projective plane." Geom. Topol. 22 (2) 1143 - 1180, 2018. https://doi.org/10.2140/gt.2018.22.1143

Information

Received: 12 July 2016; Revised: 3 May 2017; Accepted: 11 June 2017; Published: 2018
First available in Project Euclid: 1 February 2018

zbMATH: 1381.53159
MathSciNet: MR3748686
Digital Object Identifier: 10.2140/gt.2018.22.1143

Subjects:
Primary: 14J17, 53D35, 53D42

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.22 • No. 2 • 2018
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