Symplectic hypersurfaces and transversality in Gromov-Witten theory

By Kai Cieliebak, Klaus Mohnke. 2007

157 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0251626dff410b7433e1a6181093872625b537fc

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Citations in the corpus

An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2013, background): “global nite-dimensional reduction is essentially trivial (and can be done in 2Some classical style approaches using Donaldson divisors avoid nite-dimensional reductions, for instance Cieliebak{Mohnke [CM07] and Ionel{Parker [IP13]. 3 several dierent ways). Thus the entire problem is to develop a fra”
Punctured holomorphic curves and Lagrangian embeddings (2014, methodology): “The following lemma is an easy adaptation of the corresponding result in [18].”
An exact sequence for contact- and symplectic homology (2007, methodology): “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].”
Reeb orbits and the minimal discrepancy of an isolated singularity (2014, methodology): “t the conditions of Lemma 2.14 hold? The idea here is to use GromovWitten theory. Genus 0 Gromov-Witten invariants for general symplectic manifolds have now been deﬁned in many diﬀerent ways: [FO99], [CM07], [Hof11] and [LT98b]. Earlier work for special symplectic manifolds such as symplectic manifo”
A general Fredholm theory III: Fredholm functors and polyfolds (2008, methodology): “Recently, Cieliebak and Mohnke proved the genus zero case in [ 5 ] using Donaldson’s theory [7] of codimension two symplectic hypersurfaces in order to establish the crucial transversality property.”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, result): “ that hypersurface. The issues most relevant to us play out in the neighbourhood of the hypersurface. The behaviour of pseudo-holomorphic curves in such a situation is a well-explored topic, see e.g. [9, 7, 38]; our approach borrows from [33, Section 7] as well. The outcome of our discussion will be”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “One can prove that these moduli spaces are regular for generic choices of perturbation data, by essentially the same arguments as in [8] and [45, Section 9k].”