Symplectic hypersurfaces and transversality in Gromov-Witten theory

  title={Symplectic hypersurfaces and transversality in Gromov-Witten theory},
  author={Kai Cieliebak and Klaus Mohnke},
  journal={arXiv: Symplectic Geometry},
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. 
Uniruled Symplectic 4-Manifolds
This chapter includes a gentle introduction to the Gromov-Witten invariants, and then explores the relationship between these and McDuff’s results on rational/ruled surfaces, including a complete
Transversality via gauge transformations for pseudo holomorphic disks in projectivized vector bundles
We show that one can achieve transversality for lifts of holomorphic disks to a projectivized vector bundle by locally enlarging the structure group and considering the action of gauge
Symplectic capacities, unperturbed curves, and convex toric domains
We use explicit pseudoholomorphic curve techniques (without virtual perturbations) to define a sequence of symplectic capacities analogous to those defined recently by the second named author using
Absolute vs. Relative Gromov-Witten Invariants
In light of recent attempts to extend the Cieliebak-Mohnke approach for constructing Gromov-Witten invariants to positive genera, we compare the absolute and relative Gromov-Witten invariants of
A natural Gromov-Witten virtual fundamental class
We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The
Low-area Floer theory and non-displaceability
We introduce a new version of Floer theory of a non-monotone Lagrangian submanifold which only uses least area holomorphic disks with boundary on it. We use this theory to prove non-displaceability
Virtual Neighborhood Technique for Holomorphic Curve Moduli Spaces
In this paper we use the approach of Ruan and Li-Ruan to construct virtual neighborhoods and show that the Gromov-Witten invariants can be defined as an integral over top strata of virtual
Symplectic instanton homology: naturality, and maps from cobordisms
We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows
Towards a Degeneration Formula for the Gromov-Witten Invariants of Symplectic Manifolds
In this paper, we outline a project started in [7] aimed at defining Gromov-Witten (GW) invariants relative to normal crossings symplectic divisors, and GW-type invariants for normal crossings


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