Holomorphic jets in symplectic manifolds. (English) Zbl 1344.53070

The goal of this paper is to solve intersection problems that involve points on the boundary of holomorphic discs. Pointwise partial differential relations for holomorphic discs are defined here. Given a relative homotopy class, a relation, and a generic almost complex structure, the author provides the moduli space of discs that have an injective point with the structure of a smooth manifold. As an application, the local behaviour of holomorphic discs for generic almost complex structures is studied, and an inequality for the number of singularities is derived. In particular, a somewhere injective holomorphic disc is not simple in general, but in the case where the almost complex structure is chosen generically, then a somewhere injective holomorphic disc has a dense set of injective points in the interior and on the boundary, i.e., is simple and simple along the boundary. The set of non-injective points is finite if \(n \geq 3\).
The present paper extends Lazzarini’s theorem (which says that generically any nonconstant holomorphic disc is simple or multiply covered) to the remaining dimension 4. The number of double points and singularities (counted with multiplicity) is estimated in terms of topological data. An example shows here how to define Gromov-Witten-type invariants that count discs with singular points. The generic existence of immersed and embedded holomorphic curves is discussed. It is shown that, for a coordinate class of a monotone Lagrangian split torus, generically the number of non-immersed holomorphic discs is even.


53D35 Global theory of symplectic and contact manifolds
32Q65 Pseudoholomorphic curves
58A20 Jets in global analysis
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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[1] Abbas C.: Pseudoholomorphic strips in symplectisations. III. Embedding properties and compactness. J. Symplectic Geom. 2, 219-260 (2004) · Zbl 1085.53072
[2] M. Audin, Symplectic and almost complex manifolds. In: Holomorphic Curves in Symplectic Geometry, Progr. Math. 117, Birkhäuser, Basel, 1994, 41-74. · Zbl 0935.32021
[3] Barraud J.-F.: Nodal symplectic spheres in CP2 with positive self-intersection. Int. Math. Res. Not. IMRN 1999, 495-508 (1999) · Zbl 0935.32021
[4] Barraud J.-F.: Courbes pseudo-holomorphes équisingulières en dimension 4. Bull. Soc. Math. France 128, 179-206 (2000) · Zbl 0976.53037
[5] P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, Preprint, SG/0708.4221v1, 2007. · Zbl 1180.53078
[6] Cieliebak K., Mohnke K.: Symplectic hypersurfaces and transversality in Gromov-Witten theory. J. Symplectic Geom. 5, 281-356 (2007) · Zbl 1149.53052
[7] Frauenfelder U.: Gromov convergence of pseudoholomorphic disks. J. Fixed Point Theory Appl. 3, 215-271 (2008) · Zbl 1153.53057
[8] Geiges H.: h-principles and flexibility in geometry.Mem.Amer. Math.Soc. 1641, 1-58 (2003) · Zbl 1050.53001
[9] Geiges H., Zehmisch K.: Eliashberg’s proof of Cerf’s theorem. J. Topol. Anal. 2, 543-579 (2010) · Zbl 1209.57022
[10] Geiges H., Zehmisch K.: How to recognize a 4-ball when you see one. Münster J. Math. 6, 525-554 (2013) · Zbl 1300.53075
[11] Gromov M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307-347 (1985) · Zbl 0592.53025
[12] Hofer H., Zehnder E.: Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, Basel (1994) · Zbl 0805.58003
[13] S. Lang, Fundamentals of Differential Geometry. Grad. Texts in Math. 191, Springer-Verlag, New York, 1999. · Zbl 0932.53001
[14] Lazzarini L.: Existence of a somewhere injective pseudo-holomorphic disc. Geom. Funct. Anal. 10, 829-862 (2000) · Zbl 1003.32004
[15] Lazzarini L.: Relative frames on J-holomorphic curves. J. Fixed Point Theory Appl. 9, 213-256 (2011) · Zbl 1236.57035
[16] D. McDuff and D. Salamon, Introduction to Symplectic Topology. 2nd ed., Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1998. · Zbl 1066.53137
[17] D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology. Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1149.53052
[18] Oh Y.-G.: Higher jet evaluation transversality of J-holomorphic curves.J.Korean Math. Soc. 48, 341-365 (2011) · Zbl 1215.53077
[19] Oh Y.-G., Zhu K.: Embedding property of J-olomorphic curves in Calabi-Yau manifolds for generic J.Asian J. Math. 13, 323-340 (2009) · Zbl 1193.53180
[20] Oh Y.-G., Zhu K.: Floer trajectories with immersed nodes and scale-dependent gluing. J. Symplectic Geom. 9, 483-636 (2011) · Zbl 1257.53117
[21] M. Schwarz, Cohomology operations fromS1-cobordisms in Floer homology. PhD thesis, ETH-Zürich, Switzerland, 1995.
[22] J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds. In: Holomorphic Curves in Symplectic Geometry, Progr. Math. 117, Birkhäuser, Basel, 1994, 165-189.
[23] Wendl C.: Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85, 347-407 (2010) · Zbl 1207.32021
[24] C. Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry. Preprint, Version 3.1, arXiv:1011.1690, 2010.
[25] K. Zehmisch, Singularities and self-intersections of holomorphic discs. Doctoral thesis, University of Leipzig, Germany, 2008.
[26] Zehmisch K.: The annulus property of simple holomorphic discs.. J.Symplectic Geom. 11, 135-161 (2013) · Zbl 1273.32030
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