Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

Dragnev, Dragomir L.

doi:10.1002/cpa.20018

Cited by 90 publications

(171 citation statements)

References 29 publications

(46 reference statements)

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“…We then have (see [9], Proposition 1.7.1 for the dimension of the moduli spaces and [5,8] for the regularity assertions):…”

Section: Sft and Supporting Surfacesmentioning

confidence: 99%

Polarizations and symplectic isotopies

Opshtein

2013

Journal of Symplectic Geometry

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The aim of this paper is to explain a link between symplectic isotopies of open objects such as balls and flexibility properties of symplectic hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids or maximal packings of P 2 . IntroductionIn [3], Biran proved a decomposition theorem for rational Kähler manifolds which has proved useful in many situations such as symplectic packings [2, 20], Lagrangian embeddings [4] . . . This paper tries to add symplectic isotopies to this list of applications. In polarized symplectic manifolds -triples (M, ω, Σ) where ω ∈ H 2 (M, Z) and Σ is a symplectic hypersurface Poincaré-dual to a (necessarily positive) multiple kω of the symplectic form -, this decomposition result may be expressed as:Theorem 1 (Biran). In a polarized closed symplectic manifold (M, ω, Σ), there exists a zero-volume closed skeleton in M whose complement is a standard symplectic disc bundle supported by Σ.In fact, polarizations of sufficiently high degree exist on all closed (i.e., compact, without boundary) rational symplectic manifolds [7]. These manifolds split into a standard symplectic part -an explicit disc bundle over Σ -and a negligible skeleton, even isotropic in the Kähler case. Although no kind of uniqueness can be expected (see [3] for a discussion on the skeleton for instance), our next theorem explains how to construct many such decompositions, all of whose symplectic parts are isotopic. In order to state it, let us mention that both M \Σ and the complement of the zero-section L 0 in the symplectic disc bundle SDB(Σ, k) are exact symplectic manifolds, so they admit Liouville forms. In the statement below, λ 0 is a distinguished such form on SDB(Σ, k)\L 0 (see Section 1). Also, by a local embedding of (X, M ) into (Y, N ) -where X ⊂ M and Y ⊂ N -we always mean an 109

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“…We then have (see [9], Proposition 1.7.1 for the dimension of the moduli spaces and [5,8] for the regularity assertions):…”

Section: Sft and Supporting Surfacesmentioning

confidence: 99%

Polarizations and symplectic isotopies

Opshtein

2013

Journal of Symplectic Geometry

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“…I 0 I J / can be achieved by taking J to be generic inside M , while keeping it tailored. In fact, the transversality argument in [9] carries over directly to the sutured case. In particular, it suffices to perturb J arbitrarily near the periodic orbits in order to attain transversality for somewhere injective curves.…”

Section: Definition Of Sutured Contact Homologymentioning

confidence: 99%

Sutures and contact homology I

et al. 2011

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“…Lemma 1 was proved independently by Dragnev [4] using a slightly different argument. We will follow the arguments from [14], and adapt them to this situation.…”

Section: Appendix : Proof Of Lemmamentioning

confidence: 99%