Remarks on Lagrangian intersections in toric manifolds

Abreu, Miguel; Macarini, Leonardo

doi:10.1090/S0002-9947-2012-05791-6

Remarks on Lagrangian intersections in toric manifolds
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by Miguel Abreu and Leonardo Macarini PDF

Trans. Amer. Math. Soc. 365 (2013), 3851-3875 Request permission

Abstract:

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits.

References

Garrett Alston, Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions, J. Symplectic Geom. 9 (2011), no. 1, 83–106. MR 2787362
G. Alston and L. Amorim, Floer cohomology of torus fibers and real Lagrangians in Fano toric manifolds, to appear in Internat. Math. Research Notices.
Paul Biran and Octav Cornea, A Lagrangian quantum homology, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1–44. MR 2555932, DOI 10.1090/crmp/049/01
Paul Biran and Octav Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), no. 5, 2881–2989. MR 2546618, DOI 10.2140/gt.2009.13.2881
Paul Biran, Michael Entov, and Leonid Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004), no. 5, 793–802. MR 2100764, DOI 10.1142/S0219199704001525
Matthew Strom Borman, Symplectic reduction of quasi-morphisms and quasi-states, J. Symplectic Geom. 10 (2012), no. 2, 225–246. MR 2926996
M. S. Borman, Quasi-states, quasi-morphisms, and the moment map, preprint (2011), arXiv:1105.1805, to appear in Internat. Math. Research Notices.
Cheol-Hyun Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. 35 (2004), 1803–1843. MR 2057871, DOI 10.1155/S1073792804132716
Cheol-Hyun Cho, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. Geom. Phys. 58 (2008), no. 11, 1465–1476. MR 2463805, DOI 10.1016/j.geomphys.2008.06.003
C.-H. Cho and M. Poddar, Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds, preprint (2012), arXiv:1206.3994.
Cheol-Hyun Cho and Yong-Geun Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), no. 4, 773–814. MR 2282365, DOI 10.4310/AJM.2006.v10.n4.a10
D. Cox, Lectures on Toric Varieties, http://www.cs.amherst.edu/˜dac/lectures/coxcimpa. pdf.
Michael Entov and Leonid Polterovich, Rigid subsets of symplectic manifolds, Compos. Math. 145 (2009), no. 3, 773–826. MR 2507748, DOI 10.1112/S0010437X0900400X
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds. I, Duke Math. J. 151 (2010), no. 1, 23–174. MR 2573826, DOI 10.1215/00127094-2009-062
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17 (2011), no. 3, 609–711. MR 2827178, DOI 10.1007/s00029-011-0057-z
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds: A survey, preprint (2010), arXiv:1011.4044.
Dusa McDuff, Displacing Lagrangian toric fibers via probes, Low-dimensional and symplectic topology, Proc. Sympos. Pure Math., vol. 82, Amer. Math. Soc., Providence, RI, 2011, pp. 131–160. MR 2768658, DOI 10.1090/pspum/082/2768658
Dusa McDuff, The topology of toric symplectic manifolds, Geom. Topol. 15 (2011), no. 1, 145–190. MR 2776842, DOI 10.2140/gt.2011.15.145
Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993. MR 1223659, DOI 10.1002/cpa.3160460702
D. Tamarkin, Microlocal condition for non-displaceability, preprint (2008), arXiv:0809.1584.
G. Wilson and C. Woodward, Quasimap Floer cohomology and singular symplectic quotients, preprint (2011), arXiv:1105.0712.
Christopher T. Woodward, Gauged Floer theory of toric moment fibers, Geom. Funct. Anal. 21 (2011), no. 3, 680–749. MR 2810861, DOI 10.1007/s00039-011-0119-6