Journal of Symplectic Geometry 

About: Journal of Symplectic Geometry is an academic journal published by International Press of Boston, Inc.. The journal publishes majorly in the area(s): Symplectic geometry & Symplectic manifold. It has an ISSN identifier of 1527-5256. Over the lifetime, 449 publications have been published receiving 8742 citations. The journal is also known as: JSG.

Knots and Contact Geometry I: Torus Knots and the Figure Eight Knot

Abstract: We classify Legendrian torus knots and Legendrian figure eight knots in the tight contact structure on S3 up to Legendrian isotopy. A a corollary to this we also obtain the classification of transversal torus knots and transversal figure eight knots up to transversal isotopy.

Contact Toric Manifolds

Abstract: We provide a complete and self-contained classification of (compact connected) contact toric manifolds thereby finishing the work initiated by Banyaga and Molino and by Galicki and Boyer. Our motivation comes from the conjectures of Toth and Zelditch on the uniqueness of toric integrable actions on the punctured cotangent bundles on n-toru 𝕋n and of the two-sphere S2. The conjectures are equivalent to the uniqueness, up to conjugation, of maximal tori in the contactomorphism groups of the cosphere bundles of 𝕋n and S2 respectively.

Differentiable stacks and gerbes

Abstract: We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.

Gluing pseudoholomorphic curves along branched covered cylinders II

Abstract: This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U+ and U− to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential ∂ in embedded contact homology satisfies ∂2 = 0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest.

Symplectic hypersurfaces and transversality in Gromov-Witten theory

Abstract: We present a new method to prove transversality for holomorphic curves in symplectic manifolds, and show how it leads to a definition of genus zero Gromov-Witten invariants. The main idea is to introduce additional marked points that are mapped to a symplectic hypersurface of high degree in order to stabilize the domains of holomorphic maps.