2016/06/30 by Jonathan David Evans, Ivan Smith · 2 citations
Mathematics · #math.SG #math.AG #math.GT #msc:53D53 #msc:53D12
paper · pdf · doi:10.2140/gt.2018.22.1143
published as Geom. Topol. 22 (2018) 1143-1180 · 32 pages, 3 figures; v2 corrected some typos and added clarifications; v3 incorporated further corrections and comments. To appear in Geometry and Topology
arxiv created 2017/06/12 · arxiv updated 2018/03/16
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.