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Unless I'm misunderstanding what you're asking, the answer is surely no...consider for instance $\mathbb{C}P^n$ with its standard symplectic and complex structures. This admits embedded $J$-holomorph …

One of the headline consequences of Taubes' work on Seiberg-Witten theory on symplectic four- manifolds was that the standard symplectic form on $\mathbb{C}P^2$ is the unique one (up to scale, of cour …

Yes, such a Lagrangian submanifold does exist. The isotropic neighborhood theorem (see for instance p. 24 of Weinstein's Lectures on Symplectic Manifolds), together with the fact that all symplectic …

Can't you get such an obstruction by looking at periodic points? It's not difficult to show that generic Hamiltonian diffeomorphisms have only isolated fixed points, and indeed only isolated periodic …

Yes it does. Geometrically the BV operator is gotten by considering the parametrized moduli space of solutions to the Floer equation on the cylinder obtained by rotating one end of the cylinder. Thi …

Here is a formula for an explicit symplectomorphism $F$ from $T^*S^{n-1}$ to the affine quadric $\{\sum z_{j}^{2}=1\}$ in $\mathbb{C}^n$: $$ F(p,q) = \left(\frac{1+\sqrt{1+4|p|^2}}{2}\right)^{1/2} …

My impression is that for most choices of H this is a hard question. In general, the largest possible cardinality of a set of independent Poisson commuting functions on a 2n-dimensional symplectic ma …

One sort of example arises from the fact that if one starts with a Kahler form $\omega$ (which represents a class of type (1,1) in the Hodge decomposition by definition of a Kahler form), then if $\ph …

Write $X$ for the manifold in question (i.e. the symplectic blowup of the unit disc bundle in the cotangent bundle of a surface $\Sigma$). If you're interested just in monotonicity, what's relevant is …

I don't know a place where this is written up explicitly, but it's not hard as soon as one has an appropriate standard model for a neighborhood of a symplectic manifold for use in the Weinstein Neighb …

The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn' …

In case anybody is curious, there are still examples of (1) even if one replaces the requirement that the complex manifolds be nonisomorphic with the requirement that they be not even deformation equi …

Whoever told you that any embedded torus in R4 is isotopic to a Lagrangian torus was sorely mistaken. Luttinger (JDG 1995) observed the following: The manifolds obtained by doing certain Dehn-type su …

There are lots of simply-connected four-dimensional examples in Gompf's symplectic sum paper paper which are shown to be non-Kahler by virtue of violating the Noether inequality (see Theorem 6.2). Th …

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in o …