Evans, JD;
Smith, I;
(2017)
Markov numbers and Lagrangian cell complexes in the complex projective plane.
Geometry and Topology
(In press).
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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 .
| Type: | Article |
|---|---|
| Title: | Markov numbers and Lagrangian cell complexes in the complex projective plane |
| Open access status: | An open access version is available from UCL Discovery |
| Publisher version: | http://msp.org/scripts/coming.php?jpath=gt |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1559325 |
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