Symplectic hypersurfaces and transversality in Gromov-Witten theory

@article{Cieliebak2007SymplecticHA,
  title={Symplectic hypersurfaces and transversality in Gromov-Witten theory},
  author={Kai Cieliebak and Klaus Mohnke},
  journal={arXiv: Symplectic Geometry},
  year={2007}
}
We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be achieved by perturbing the almost complex structure. 
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125 Citations
Highly Influential Citations
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Background Citations
30
Methods Citations
38
Results Citations
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125 Citations

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References

SHOWING 1-10 OF 25 REFERENCES
Symplectic hypersurfaces in the complement of an isotropic submanifold
Abstract. Using Donaldson's approximately holomorphic techniques, we construct symplectic hypersurfaces lying in the complement of any given compact isotropic submanifold of a compact symplectic…
Relative Gromov-Witten invariants
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of…
Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
We construct Gromov-Witten invariants of general symplectic manifolds.
Transversality in elliptic Morse theory for the symplectic action
Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the denition of symplectic…
The symplectic sum formula for Gromov–Witten invariants
In the symplectic category there is a 'connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula…
Virtual Moduli Cycles and Gromov-Witten Invariants of Noncompact Symplectic Manifolds
This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also…
Introduction to Symplectic Field Theory
We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds…
The unregularized gradient flow of the symplectic action
The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this…
Asymptotically Holomorphic Families of Symplectic Submanifolds
Abstract. We construct a wide range of symplectic submanifolds in a compact symplectic manifold as the zero sets of asymptotically holomorphic sections of vector bundles obtained by tensoring an…
Symplectic Gromov-Witten Invariants
The theory of Gromov-Witten invariants has its origins in Gromov’s pioneering work. Encouraged by conjectures coming from physicists, it took a while until a rigorous mathematical foundation was…
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