Markov numbers and Lagrangian cell complexes in the complex projective plane

Evans, Jonathan David and Smith, Ivan (2018) Markov numbers and Lagrangian cell complexes in the complex projective plane. Geometry and Topology, 22 (2). pp. 1143-1180. ISSN 1364-0380

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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.

Item Type:
Journal Article
Journal or Publication Title:
Geometry and Topology
Subjects:
ID Code:
132430
Deposited By:
Deposited On:
02 Apr 2019 09:15
Refereed?:
Yes
Published?:
Published
Last Modified:
24 Nov 2020 07:21