Geodesic
Displacing Lagrangian toric fibers by extended probes
Abreu, Miguel1 ; Borman, Matthew Strom2 ; McDuff, Dusa3
1Centro de Análise Mathemática, Geometria e Sistemas Dinâmicos, Departamento de Mathemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
2Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
3Mathematics Department, Barnard College and Columbia University, New York, NY 10027, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 687-752
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In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space ℙ(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

DOI : 10.2140/agt.2014.14.687
Classification : 53D12, 14M25, 53D40
Keywords: symplectic manifolds, Lagrangian tori, nondisplaceable, toric manifolds

Abreu, Miguel  1   ; Borman, Matthew Strom  2   ; McDuff, Dusa  3

1 Centro de Análise Mathemática, Geometria e Sistemas Dinâmicos, Departamento de Mathemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
2 Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
3 Mathematics Department, Barnard College and Columbia University, New York, NY 10027, USA 
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Abreu, Miguel; Borman, Matthew Strom; McDuff, Dusa. Displacing Lagrangian toric fibers by extended probes. Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 687-752. doi: 10.2140/agt.2014.14.687

[1] M Abreu, L Macarini, Remarks on Lagrangian intersections in toric manifolds, Trans. Amer. Math. Soc. 365 (2013) 3851

[2] P Biran, O Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009) 2881

[3] P Biran, M Entov, L Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004) 793

[4] M S Borman, Quasi-states, quasi-morphisms, and the moment map, Int. Math. Res. Not. 2013 (2013) 2497

[5] D M J Calderbank, M A Singer, Einstein metrics and complex singularities, Invent. Math. 156 (2004) 405

[6] Y Chekanov, F Schlenk, Notes on monotone Lagrangian twist tori, Electron. Res. Announc. Math. Sci. 17 (2010) 104

[7] C H Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. 2004 (2004) 1803

[8] C H Cho, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. Geom. Phys. 58 (2008) 1465

[9] C H Cho, Y G Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006) 773

[10] C H Cho, M Poddar, Holomorphic orbidiscs and Lagrangian Floer cohomology of compact toric orbifolds

[11] T Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988) 315

[12] M Entov, L Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv. 81 (2006) 75

[13] M Entov, L Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009) 773

[14] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds: Survey

[15] K Fukaya, Y G Oh, H Ohta, K Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory

[16] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 (2010) 23

[17] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian Floer theory on compact toric manifolds II: Bulk deformations, Selecta Math. 17 (2011) 609

[18] Y Karshon, E Lerman, Non-compact symplectic toric manifolds

[19] E Lerman, E Meinrenken, S Tolman, C Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998) 245

[20] E Lerman, S Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997) 4201

[21] D Mcduff, Displacing Lagrangian toric fibers via probes, from: "Low-dimensional and symplectic topology" (editor M Usher), Proc. Sympos. Pure Math. 82, Amer. Math. Soc. (2011) 131

[22] D Mcduff, S Tolman, Polytopes with mass linear functions, I, Int. Math. Res. Not. 2010 (2010) 1506

[23] P Orlik, F Raymond, Actions of the torus on $4$–manifolds, I, Trans. Amer. Math. Soc. 152 (1970) 531

[24] G Wilson, C T Woodward, Quasimap Floer cohomology for varying symplectic quotients, Canad. J. Math. 65 (2013) 467

[25] C T Woodward, Gauged Floer theory of toric moment fibers, Geom. Funct. Anal. 21 (2011) 680

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