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Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ... 0answers 73 views Compactly supported symplectomorphisms of$D^2$I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of$D^2 \times D^2$being contractible. Consider the dimensional disk$D^2 \...
362 views

Example of two exotic closed 4-manifolds s.t. SW(X)=0

I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic. So far in the literature I've only found examples which ...
97 views

Every symplectic submanifold is J-holomorphic

I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$. The way I am ...
149 views

A special non vanishing vector field on odd dimensional compact manifolds

Edit: According to the comment of Michael Albanese we revise the question. Assume that $n$ is an odd integer and $M\subset \mathbb{R}^{2n}$ is a compact orientable $n$ dimensional submanifold. Does ...
172 views

Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\... 0answers 69 views Transitivity of Diff on the space of embeddings of balls Given 2 symplectic embeddings$g_0$and$g_1$of a 4-ball of radius$r \leq 1$into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from$\mathbb{R}^4$), does there exist ... 2answers 322 views Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces I apologize in advance if this question has an obvious answer. Let$(M,g)$be a Riemannian manifold. Then the tangent bundle$TM$carries a natural symplectic structure$\omega_g$. In fact$\omega_g$... 1answer 642 views What can we say about the Cartesian product of a manifold with its exotic copy? Let$M$be a smooth oriented manifold, and let$M^E$be an exotic copy, i.e homeomorphic but not diffeomorphic to$M$. Is it true that$M\times M$is diffeomorphic to$M\times M^E$? I am ... 1answer 151 views What is symplectic cut of a 4-ball? Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the$\mathbb{S}^1$orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. ... 1answer 125 views Existence of isotopy preserving the action Let$\gamma_1$and$\gamma_2$be simple closed curves in$R^4.$Let$\lambda= x_1 dy_1+ x_2dy_2.$Suppose that$\int_{\gamma_1} \lambda= \int_{\gamma_2} \lambda.$I am looking for a reference for ... 1answer 295 views When is a Divergence-Free Vector Field on the Tangent Bundle of a Riemannian Manifold Hamiltonian? (Reposted from https://math.stackexchange.com/questions/2589600/when-is-a-divergence-free-vector-field-on-the-tangent-bundle-of-a-riemannian-man) Starting with a closed, connected Riemannian manifold$...
111 views

Hamiltonian Group action with infinitely many stabiliser types

What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types? Infinitely many stabiliser types means that ...
254 views

Every contractible smooth loop has a neighbourhood with $H^2=0$

Let $c: S^1 \to M$ be a smooth contractible loop (not necesarily an embedding, or even an immersion) on the connected, compact symplectic manifold $(M,\omega)$ (if this helps somehow, $c$ is a $1$-...
500 views

$(M,\omega)$ not symplectomorphic to $(M,-\omega)$

Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$. In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...

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