Markov numbers and Lagrangian cell complexes in the complex projective plane

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 $(1 ∕ p^{2}) (p q - 1, 1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into ${ℂ ℙ}^{2}$ 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 $N \leq 3$ 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 ${ℂ ℙ}^{2}$ .

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Keywords

symplectic four-manifolds and orbifolds, Markov numbers, Wahl singularities, vanishing cycles

Mathematical Subject Classification 2010

Primary: 14J17, 53D35, 53D42

References

Publication

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

Authors

Jonathan David Evans
Department of Mathematics University College London London United Kingdom

Ivan Smith
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Volume 22, issue 2 (2018)

Jonathan David Evans and Ivan Smith

Abstract