The Gromov width of complex Grassmannians

Karshon, Yael¹ ; Tolman, Susan²

¹Department of Mathematics, the University of Toronto, Toronto, Ontario M5S 3G3, Canada
²Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green St, Urbana IL 61801, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 911-922

Résumé

We show that the Gromov width of the Grassmannian of complex k–planes in ℂn is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general results. For example, if a compact manifold N with an integral symplectic form ω admits a Hamiltonian circle action with a fixed point p such that all the isotropy weights at p are equal to one, then the Gromov width of (N,ω) is at least one. We use holomorphic techniques to prove the upper bound.

Détail

DOI : 10.2140/agt.2005.5.911

Keywords: Gromov width, Moser's method, symplectic embedding, complex Grassmannian, moment map

Bibliographie

[1] , Holomorphic curves in symplectic geometry, Progress in Mathematics 117, Birkhäuser Verlag (1994)

[2] M F Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982) 1

[3] A Bertram, G Daskalopoulos, R Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996) 529

[4] P Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997) 420

[5] P Biran, A stability property of symplectic packing, Invent. Math. 136 (1999) 123

[6] P Biran, From symplectic packing to algebraic geometry and back, from: "European Congress of Mathematics, Vol II (Barcelona, 2000)", Progr. Math. 202, Birkhäuser (2001) 507

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

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

[9] V Guillemin, S Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982) 491

[10] Y Karshon, S Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001) 4831

[11] B Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices (1995) 1

[12] J L Koszul, Sur certains groupes de transformations de Lie, from: "Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953", Centre National de la Recherche Scientifique (1953) 137

[13] F Lalonde, D Mcduff, The geometry of symplectic energy, Ann. of Math. $(2)$ 141 (1995) 349

[14] G Lu, Gromov–Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1

[15] D Mcduff, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 23 (1990) 311

[16] D Mcduff, L Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994) 405

[17] D Mcduff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994)

[18] B Siebert, G Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997) 679

[19] L Traynor, Symplectic packing constructions, J. Differential Geom. 42 (1995) 411

[20] A Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971)