Symplectic hypersurfaces and transversality in Gromov-Witten theory

Cieliebak, Kai; Mohnke, Klaus

doi:10.4310/jsg.2007.v5.n3.a2

Cited by 100 publications

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“…We expect these technical assumptions to be completely removed using the new ongoing approach to transversality by Cieliebak and Mohnke (see [13] for the symplectic case), or using the polyfold theory developed by Hofer, Wysocki and Zehnder [18,21]. We give below the list of examples known to us in which both these conditions are satisfied.…”

Section: One Can Associate a Sign ǫ(F ) To Each Element [F ] ∈ Mmentioning

confidence: 99%

“…In order for the linearized contact differential to be well-defined on i −1 (a) we need that the almost complex structure J on W satisfies the following conditions. (A) J is regular for holomorphic planes belonging to moduli spaces M A (γ ′ , ∅; J) of virtual dimension ≤ 0; (B a ) J ∞ is regular for punctured holomorphic cylinders asymptotic at ±∞ to closed Reeb orbits in i −1 (a), belonging to moduli spaces of virtual dimension ≤ 2, and which are asymptotic at the punctures to closed Reeb orbits γ ′ such that M A (γ ′ , ∅; J) = ∅ and has virtual dimension 0.We expect these technical assumptions to be completely removed using the new ongoing approach to transversality by Cieliebak and Mohnke (see [13] for the symplectic case), or using the polyfold theory developed by Hofer, Wysocki and Zehnder [18,21]. We give below the list of examples known to us in which both these conditions are satisfied.…”

mentioning

confidence: 99%

“…We refer to Section 3.1, Remark 9 for a discussion of these regularity assumptions. We expect this technical assumption to be completely removed using the new ongoing approach to transversality by Cieliebak and Mohnke (see [13] for the symplectic case), or using the polyfold theory developed by Hofer Wysocki and Zehnder [18,21]. …”

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confidence: 99%

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An exact sequence for contact- and symplectic homology

2008

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Abstract. A symplectic manifold W with contact type boundary M = ∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M ). We establish a Gysin-type exact sequence in which the symplectic homology SH(W ) of W maps to HC(M ), which in turn maps to HC(M ), by a map of degree −2, which then maps to SH(W ). Furthermore, we give a description of the degree −2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M . ContentsSuch an X is called a Liouville vector field. The 1-form λ := (ι X ω)| M is a contact form on M . We denote by ξ the contact distribution defined by λ and we call (W, ω) a filling of (M, ξ).We assume throughout the paper that (W, ω) satisfies the conditionT 2 f * ω = 0 for all smooth f :where T 2 is the 2-torus. This condition guarantees that the energy of a Floer trajectory (for a definition, see for example [7, Section 2]) does not depend on its homology class, but only on its endpoints. Our main class of examples is provided by exact symplectic forms.Theorem 1 ties together the symplectic homology groups of (W, ω) and the linearized contact homology groups of (M, ξ). Both these invariants encode algebraically the dynamics of the same vector field, the Reeb vector field R λ defined by ker ω| M = R λ and λ(R λ ) = 1. But their natures are quite different: the former belongs to the realm of Floer theory [17,32], whereas the latter belongs to the realm of symplectic field theory (SFT) [16]. Our result can be read as a way to make symplectic homology fit into SFT.Let us introduce some relevant notation. Given a free homotopy class a of loops in W we denote by SH a * (W, ω) the symplectic homology groups of (W, ω) in the homotopy class a. The free homotopy class of the constant loop will be denoted by 0. We also denote by SH + * (W, ω) the symplectic homology groups in the trivial homotopy class truncated at a small positive value of the action functional. We refer to Section 2 for the definitions.Let i : M ֒→ W be the inclusion. Given a free homotopy class a of loops in W we denote by i −1 (a) the set of free homotopy classes in M which are mapped to a via i, and we use the convention i −1 (+) := i −1 (0). We denote by HC i −1 (a) * (M, ξ) the linearized contact homology groups of (M, ξ) based on closed Reeb orbits whose free homotopy class belongs to i −1 (a). We refer to Section 3.1 for the definition.Both SH a * (W, ω) and HC i −1 (a) * (M, ξ) are defined over the Novikov ring Λ ω with Q-coefficients consisting of formal combinations λ := A∈H2(W ;Z) λ A e A , λ A ∈ Q such that #{A|λ A = 0, ω(A) ≤ c} < ∞ for all c > 0. The multiplication in Λ ω is given by the convolution product.We assume the existence of an almost complex structure J such that linearized contact homology is defined. This means that J needs to be regular for rigid holomorphic planes in the symplectic completion of W , as well as for rational holomorphic curves with one positive puncture in the symplectization of M satisfyin...

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Section: One Can Associate a Sign ǫ(F ) To Each Element [F ] ∈ Mmentioning

confidence: 99%

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confidence: 99%

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An exact sequence for contact- and symplectic homology

2008

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“…Using these explicitly defined almost complex structures, we fix a tame almost complex structure J ∞ on (X \ L, ω) coinciding with J cyl in the coordinates near L given by Ψ above. We then consider a family J τ , τ ≥ 0, of tame almost complex structures on (X, ω) determined uniquely by the following properties: The compactness theorem [25], [31], [6], [12] shows that sequences of J τ -holomorphic spheres in X have convergent subsequences to split pseudoholomorphic spheres in the following sense as τ → ∞ . [12]).…”

Section: Split Pseudoholomorphic Curvesmentioning

confidence: 99%

“…We then consider a family J τ , τ ≥ 0, of tame almost complex structures on (X, ω) determined uniquely by the following properties: The compactness theorem [25], [31], [6], [12] shows that sequences of J τ -holomorphic spheres in X have convergent subsequences to split pseudoholomorphic spheres in the following sense as τ → ∞ . [12]). Consider a sequence u i of J τ i -holomorphic spheres in (X, ω) in a fixed homology class A ∈ H 2 (X), where τ i → +∞ as i → ∞.…”

Section: Split Pseudoholomorphic Curvesmentioning

confidence: 99%

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

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Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

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Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

Gerig

Wendl

2016

Comm Pure Appl Math

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We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index 0. A corollary is that the Gromov‐Witten invariants (without descendants) of symplectic 4‐manifolds can always be computed as a signed and weighted count of honest J‐holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homology class. The transversality proof is based on an analytic perturbation technique, originally due to Taubes.© 2016 Wiley Periodicals, Inc.

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Symplectic hypersurfaces and transversality in Gromov-Witten theory

Cited by 100 publications

References 17 publications

An exact sequence for contact- and symplectic homology

An exact sequence for contact- and symplectic homology

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

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