On certain Lagrangian submanifolds of S2 ×S2 and ℂPn

Oakley, Joel¹ ; Usher, Michael²

¹Department of Mathematics, Belhaven University, Jackson, MS 39202, USA
²Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 149-209

Résumé

We consider various constructions of monotone Lagrangian submanifolds of ℂ Pn, S2 × S2, and quadric hypersurfaces of ℂ Pn. In S2 × S2 and ℂ P2 we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of ℂ P2 is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of ℂ Pn which can be understood either in terms of the geodesic flow on T∗Sn or in terms of the Biran circle bundle construction. Unlike previously known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.

Détail

DOI : 10.2140/agt.2016.16.149

Classification : 53D12
Keywords: Lagrangian submanifolds, Hamiltonian displaceability

Bibliographie

[1] P Albers, U Frauenfelder, A nondisplaceable Lagrangian torus in T∗S2, Comm. Pure Appl. Math. 61 (2008) 1046

[2] D Auroux, Mirror symmetry and T–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. 1 (2007) 51

[3] P Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407

[4] P Biran, Lagrangian non-intersections, Geom. Funct. Anal. 16 (2006) 279

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

[6] P Biran, O Cornea, Lagrangian topology and enumerative geometry, Geom. Topol. 16 (2012) 963

[7] P Biran, M Khanevsky, A Floer–Gysin exact sequence for Lagrangian submanifolds, Comment. Math. Helv. 88 (2013) 899

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

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

[10] Y Eliashberg, L Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, preprint (2010)

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

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

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

[14] K Fukaya, Y G Oh, H Ohta, K Ono, Toric degeneration and nondisplaceable Lagrangian tori in S2 × S2, Int. Math. Res. Not. 2012 (2012) 2942

[15] A Gadbled, On exotic monotone Lagrangian tori in CP2 and S2 × S2, J. Symplectic Geom. 11 (2013) 343

[16] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307

[17] M Khovanov, P Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002) 203

[18] E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247

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

[20] 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

[21] Y G Oh, Addendum to : “Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, I” [Comm. Pure Appl. Math. 46 (1993) 949–993], Comm. Pure Appl. Math. 48 (1995) 1299

[22] L Polterovich, The geometry of the group of symplectic diffeomorphisms, Birkhäuser (2001)

[23] M Poźniak, Floer homology, Novikov rings and clean intersections, from: "Northern California Symplectic Geometry Seminar" (editors Y Eliashberg, D Fuchs, T Ratiu, A Weinstein), Amer. Math. Soc. Transl. 196, Amer. Math. Soc. (1999) 119

[24] R Vianna, Infinitely many exotic monotone Lagrangian tori in CP2, preprint (2014)

[25] R Vianna, On exotic Lagrangian tori in CP2, Geom. Topol. 18 (2014) 2419

[26] W Wu, On an exotic Lagrangian torus in CP2, Compos. Math. 151 (2015) 1372