Related papers: Cohomologous symplectic forms with different Gromo…

We show that the Gromov width of the Grassmannian of complex k-planes in C^n is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general…

In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the…

In the present paper we study the variation of the dimensions $h_k$ of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all…

We construct a uniformly bounded symplectic structure on $S^2 \times \mathbb{R}^4$ admitting embeddings by arbitrarily large balls. This provides a counterexample to a recent conjecture of Savelyev. We then prove the conjecture holds for a…

This paper explains an application of Gromov's h-principle to prove the existence, on any orientable 4-manifold, of a folded symplectic form. That is a closed 2-form which is symplectic except on a separating hypersurface where the form…

Let $(X,\om)$ be a symplectic manifold and $L$ be a Lagrangian submanifold diffeomorphic to $S^n$, $\R\P^n$, or a Lens space of a certain type. Using the symplectic cut and symplectic sum constructions, we express the open Gromov-Witten…

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that…

We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic…

In this note, genus zero Gromov-Witten invariants are reviewed and then applied in some examples of dimension four and six. It is also proved that the use of genus zero Gromov-Witten invariants in the class of embedded $J$-holomorphic…

Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the…

Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a…

Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of $X$ and $Y$. This formula has…

We examine how symplectic cohomology may be used as an invariant on symplectic structures, and investigate the non-uniqueness of these structures on Liouville domains, a field which has seen much development in the past decade. Notably, we…

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective…

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/\mu_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold…

We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of…

Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area $R$, and let $\Sigma_g(L)$ be a closed symplectic surface of genus $g$ and area $L$. We prove that there is a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \times…

We find an upper bound for the Gromov width of coadjoint orbits of U(n) with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one…

We initiate here the study of Gromov-Witten theory of locally conformally symplectic manifolds or $\lcs$ manifolds, $\lcsm$'s for short, which are a natural generalization of both contact and symplectic manifolds. We find that the main new…

We describe the deformation cohomology of a symplectic groupoid, and use it to study deformations via Moser path methods, proving a symplectic groupoid version of the Moser Theorem. Our construction uses the deformation cohomologies of Lie…