TY - JOUR

T1 - Markov numbers and Lagrangian cell complexes in the complex projective plane

AU - Evans, Jonathan David

AU - Smith, Ivan

PY - 2018/1/16

Y1 - 2018/1/16

N2 - 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.

AB - 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.

KW - symplectic four-manifolds and orbifolds

KW - markov numbers

KW - wahl singularities

KW - vanishing cycles

U2 - 10.2140/gt.2018.22.1143

DO - 10.2140/gt.2018.22.1143

M3 - Journal article

VL - 22

SP - 1143

EP - 1180

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1364-0380

IS - 2

ER -