Rigidity and uniruling for Lagrangian submanifolds

Biran, Paul¹ ; Cornea, Octav²

¹ School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
² Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada

Geometry & topology, Tome 13 (2009) no. 5, pp. 2881-2989.

Résumé

This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

Détail

DOI : 10.2140/gt.2009.13.2881

Keywords: symplectic manifold, Lagrangian submanifold

Bibliographie

[1] P Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology

[2] P Albers, On the extrinsic topology of Lagrangian submanifolds, Int. Math. Res. Not. (2005) 2341

[3] M Audin, F Lalonde, L Polterovich, Symplectic rigidity: Lagrangian submanifolds, from: "Holomorphic curves in symplectic geometry" (editors M Audin, J Lafontaine), Progr. Math. 117, Birkhäuser (1994) 271

[4] J F Barraud, O Cornea, Homotopic dynamics in symplectic topology, from: "Morse theoretic methods in nonlinear analysis and in symplectic topology" (editors P Biran, O Cornea, F Lalonde), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 109

[5] J F Barraud, O Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. $(2)$ 166 (2007) 657

[6] A Beauville, Quantum cohomology of complete intersections, Mat. Fiz. Anal. Geom. 2 (1995) 384

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

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

[9] P Biran, K Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv. 76 (2001) 712

[10] P Biran, K Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel J. Math. 127 (2002) 221

[11] P Biran, O Cornea, in preparation

[12] P Biran, O Cornea, Quantum structures for Lagrangian submanifolds

[13] P Biran, O Cornea, A Lagrangian quantum homology, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes, Amer. Math. Soc. (2009)

[14] L Blechman, L Polterovich, private communication

[15] L Buhovsky, Multiplicative structures in Lagrangian Floer homology

[16] L Buhovsky, One explicit construction of a relative packing

[17] L Buhovsky, Homology of Lagrangian submanifolds in cotangent bundles, Israel J. Math. 143 (2004) 181

[18] Y V Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms, Math. Z. 223 (1996) 547

[19] Y V Chekanov, F Schlenk, Monotone Lagrangian tori in $\mathbb{R}^{2n}$, $\mathbb{C}P^n$ and products of spheres, in preparation

[20] R Chiang, New Lagrangian submanifolds of $\mathbb{CP}^n$, Int. Math. Res. Not. (2004) 2437

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

[22] C H Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005) 613

[23] 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

[24] O Cornea, F Lalonde, Cluster homology

[25] O Cornea, F Lalonde, Cluster homology: an overview of the construction and results, Electron. Res. Announc. Amer. Math. Soc. 12 (2006) 1

[26] O Cornea, A Ranicki, Rigidity and gluing for Morse and Novikov complexes, J. Eur. Math. Soc. $($JEMS$)$ 5 (2003) 343

[27] Y Eliashberg, Topological characterization of Stein manifolds of dimension $\gt 2$, Internat. J. Math. 1 (1990) 29

[28] Y Eliashberg, M Gromov, Convex symplectic manifolds, from: "Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989)" (editors E Bedford, J P D’Angelo, R E Greene, S G Krantz), Proc. Sympos. Pure Math. 52, Amer. Math. Soc. (1991) 135

[29] Y Eliashberg, L Polterovich, The problem of Lagrangian knots in four-manifolds, from: "Geometric topology (Athens, GA, 1993)" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 313

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

[31] U Frauenfelder, Gromov convergence of pseudoholomorphic disks, J. Fixed Point Theory Appl. 3 (2008) 215

[32] K Fukaya, Morse homotopy and its quantization, from: "Geometric topology (Athens, GA, 1993)" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 409

[33] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory – anomaly and obstruction, Preprint

[34] P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Math., Wiley-Interscience (1978)

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

[36] D Kwon, Y G Oh, Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition, Comm. Anal. Geom. 8 (2000) 31

[37] L Lazzarini, Relative frames on $J$–holomorphic curves, to appear in J. Fixed Point Theory Appl.

[38] L Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal. 10 (2000) 829

[39] D Mcduff, Hamiltonian $S^{1}$–manifolds are uniruled

[40] D Mcduff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc. (2004)

[41] Y G Oh, Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group

[42] Y G Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993) 949

[43] Y G Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices (1996) 305

[44] Y G Oh, Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds, from: "Contact and symplectic geometry (Cambridge, 1994)" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 201

[45] Y G Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, from: "The breadth of symplectic and Poisson geometry" (editors J E Marsden, T S Ratiu), Progr. Math. 232, Birkhäuser (2005) 525

[46] Y G Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Math. J. 130 (2005) 199

[47] Y G Oh, Lectures on Floer theory and spectral invariants of Hamiltonian flows, from: "Morse theoretic methods in nonlinear analysis and in symplectic topology" (editors P Biran, O Cornea, F Lalonde), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 321

[48] S Piunikhin, D Salamon, M Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, from: "Contact and symplectic geometry (Cambridge, 1994)" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 171

[49] M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000) 419

[50] P Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046

[51] P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103

[52] P Seidel, Exact Lagrangian submanifolds in $T^{*}S^n$ and the graded Kronecker quiver, from: "Different faces of geometry" (editors S Donaldson, Y Eliashberg, M Gromov), Int. Math. Ser. 3, Kluwer/Plenum (2004) 349

[53] C Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992) 685