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The Gromov width of 4-dimensional tori. (English) Zbl 1277.57024

A symplectic manifold is a \(2n\)-dimensional manifold \(M\) together with a closed 2-form \(\omega\) such that \(\omega^n\) is non-zero everywhere. For example, \(\omega_0:=dx_1\wedge dy_1+dx_2\wedge dy_2\) is a symplectic structure on \(\mathbb{R}^4\). Other examples are given by the quotient of \(({\mathbb R}^4,\omega_0)\) by a suitable lattice \(\Lambda\). Such a quotient manifold is necessarily a 4-torus, and the resulting symplectic form is called a linear symplectic form on a 4-torus. It is not known whether every symplectic form on a 4-torus is symplectomorphic to such a form.
Given \(a>0\) we consider the open ball of capacity \(a\), i.e. we consider \[ B^4(a):=\{ z\in \mathbb{C}^2\,|\, \pi(|z_1|^2+|z_2|^2|)<a\} \] as a subset of the symplectic space \(({\mathbb R}^4,\omega_0)\). The ball filling number of a finite volume symplectic manifold \((M,\omega)\) is defined as \[ p(M,\omega):=\text{sup} \frac{ \text{Vol}(B^4(a))}{\text{Vol}(M,\omega)}, \] where the supremum is taken over all balls \(B^4(a)\) that symplectically embed into \((M,\omega)\). If \(p(M,\omega)<1\), then one says that there is a filling obstruction, while if \(p(M,\omega)=1\), one says that \((M,\omega)\) admits a full filling by one ball.
There are symplectic manifolds which have a filling obstruction. But the main theorem of the paper says that a 4-torus with a linear symplectic form admits a full filling by one ball.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
57R40 Embeddings in differential topology
32J27 Compact Kähler manifolds: generalizations, classification
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[1] T Bauer, Seshadri constants and periods of polarized abelian varieties, Math. Ann. 312 (1998) 607 · Zbl 0933.14025 · doi:10.1007/s002080050238
[2] T Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (1999) 547 · Zbl 0955.14005 · doi:10.1007/s002080050272
[3] T Bauer, T Szemberg, Local positivity of principally polarized abelian threefolds, J. Reine Angew. Math. 531 (2001) 191 · Zbl 0964.14005 · doi:10.1515/crll.2001.014
[4] P Biran, Symplectic packing in dimension \(4\), Geom. Funct. Anal. 7 (1997) 420 · Zbl 0892.53022 · doi:10.1007/s000390050014
[5] P Biran, A stability property of symplectic packing, Invent. Math. 136 (1999) 123 · Zbl 0930.53052 · doi:10.1007/s002220050306
[6] P Biran, K Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv. 76 (2001) 712 · Zbl 1001.53057 · doi:10.1007/s00014-001-8326-7
[7] C Birkenhake, H Lange, Complex abelian varieties, Grundl. Math. Wissen. 302, Springer (2004) · Zbl 1056.14063
[8] N Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999) 287 · Zbl 0926.32025 · doi:10.5802/aif.1674
[9] P Cascini, D Panov, Symplectic generic complex structures on four-manifolds with \(b_+ = 1\), J. Symplectic Geom. 10 (2012) 493 · Zbl 1261.53069 · doi:10.4310/JSG.2012.v10.n4.a1
[10] F Catanese, K Oguiso, T Peternell, On volume-preserving complex structures on real tori, Kyoto J. Math. 50 (2010) 753 · Zbl 1231.32011 · doi:10.1215/0023608X-2010-013
[11] O Debarre, Higher-dimensional algebraic geometry, Springer (2001) · Zbl 0978.14001
[12] T Dr\uaghici, The Kähler cone versus the symplectic cone, Bull. Math. Soc. Sci. Math. Roumanie 42 (1999) 41 · Zbl 0953.53043
[13] I Ekeland, H Hofer, Symplectic topology and Hamiltonian dynamics, II, Math. Z. 203 (1990) 553 · Zbl 0729.53039 · doi:10.1007/BF02570756
[14] G Elencwajg, O Forster, Vector bundles on manifolds without divisors and a theorem on deformations, Ann. Inst. Fourier (Grenoble) 32 (1982) 25 · Zbl 0488.32012 · doi:10.5802/aif.893
[15] C Fefferman, D H Phong, The uncertainty principle and sharp G\rarding inequalities, Comm. Pure Appl. Math. 34 (1981) 285 · Zbl 0458.35099 · doi:10.1002/cpa.3160340302
[16] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 · Zbl 0592.53025 · doi:10.1007/BF01388806
[17] H Hofer, E Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser (1994) · Zbl 0805.58003 · doi:10.1007/978-3-0348-8540-9
[18] M Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011) 231 · Zbl 1238.53061
[19] M Y Jiang, Symplectic embeddings from \(\mathbfR^{2n}\) into some manifolds, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 53 · Zbl 0951.53052 · doi:10.1017/S0308210500000044
[20] Y Karshon, S Tolman, The Gromov width of complex Grassmannians, Algebr. Geom. Topol. 5 (2005) 911 · Zbl 1092.53062 · doi:10.2140/agt.2005.5.911
[21] J L Lagrange, Solution d’un problème d’arithmétique (editor J A Serret), Georg Olms Verlag (1973) 671
[22] F Lalonde, D McDuff, The geometry of symplectic energy, Ann. of Math. 141 (1995) 349 · Zbl 0829.53025 · doi:10.2307/2118524
[23] F Lalonde, D McDuff, Hofer’s \(L^\infty\)-geometry: Energy and stability of Hamiltonian flows, II, Invent. Math. 122 (1995) 35 · Zbl 0844.58021 · doi:10.1007/BF01231438
[24] F Lalonde, D McDuff, The classification of ruled symplectic \(4\)-manifolds, Math. Res. Lett. 3 (1996) 769 · Zbl 0874.57019 · doi:10.4310/MRL.1996.v3.n6.a5
[25] A Lamari, Le cône kählérien d’une surface, J. Math. Pures Appl. 78 (1999) 249 · Zbl 0941.32007 · doi:10.1016/S0021-7824(98)00005-1
[26] R Lazarsfeld, Lengths of periods and Seshadri constants of abelian varieties, Math. Res. Lett. 3 (1996) 439 · Zbl 0890.14025 · doi:10.4310/MRL.1996.v3.n4.a1
[27] R Lazarsfeld, Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, Ergeb. Math. Grenzgeb. 48, Springer (2004) · Zbl 1093.14501 · doi:10.1007/978-3-642-18808-4
[28] T J Li, M Usher, Symplectic forms and surfaces of negative square, J. Symplectic Geom. 4 (2006) 71 · Zbl 1120.53052 · doi:10.4310/JSG.2006.v4.n1.a4
[29] G Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1 · Zbl 1133.53059 · doi:10.1007/BF02773823
[30] D McDuff, Blow ups and symplectic embeddings in dimension \(4\), Topology 30 (1991) 409 · Zbl 0731.53035 · doi:10.1016/0040-9383(91)90021-U
[31] D McDuff, From symplectic deformation to isotopy (editor R J Stern), First Int. Press Lect. Ser. 1, Int. Press (1998) 85 · Zbl 0928.57018
[32] D McDuff, Geometric variants of the Hofer norm, J. Symplectic Geom. 1 (2002) 197 · Zbl 1037.37033 · doi:10.4310/JSG.2001.v1.n2.a2
[33] D McDuff, L Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994) 405 · Zbl 0833.53028 · doi:10.1007/BF01231766
[34] D McDuff, D Salamon, Introduction to symplectic topology, The Clarendon Press (1998) · Zbl 0844.58029
[35] D McDuff, J Slimowitz, Hofer-Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol. 5 (2001) 799 · Zbl 1002.57056 · doi:10.2140/gt.2001.5.799
[36] F Schlenk, Embedding problems in symplectic geometry, de Gruyter Expositions in Mathematics 40, de Gruyter (2005) · Zbl 1073.53117 · doi:10.1515/9783110199697
[37] A Steffens, Remarks on Seshadri constants, Math. Z. 227 (1998) 505 · Zbl 0927.14002 · doi:10.1007/PL00004388
[38] C H Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995) 221 · Zbl 0854.57020 · doi:10.4310/MRL.1995.v2.n2.a10
[39] C H Taubes, Seiberg Witten and Gromov invariants for symplectic \(4\)-manifolds, First Int. Press Lect. Series 2, Int. Press (2000) · Zbl 0967.57001
[40] L Traynor, Symplectic packing constructions, J. Differential Geom. 41 (1995) 735 · Zbl 0830.52011
[41] C T C Wall, Diffeomorphisms of \(4\)-manifolds, J. London Math. Soc. 39 (1964) 131 · Zbl 0121.18101 · doi:10.1112/jlms/s1-39.1.131
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