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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.

MSC:

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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