# Questions tagged [symplectic-topology]

The symplectic-topology tag has no usage guidance.

155
questions

**1**

vote

**1**answer

39 views

### Lagrangian torus fibrations and Arnol'd-Liouville theorem

Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of ...

**4**

votes

**1**answer

96 views

### Orientable surface bundle

Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why?
What about the particular case that $M$ is a connected compact 4-manifold?

**2**

votes

**0**answers

70 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 \...

**6**

votes

**0**answers

112 views

### Exponential decay of a $J$-holomorphic map on a long cylinder

Suppose $(X,\omega,J)$ is a closed symplectic manifold with a compatible almost complex structure. The fact below follows from McDuff-Salamon's book on $J$-holomorphic curves (specifically, Lemma 4.7....

**1**

vote

**0**answers

111 views

### The norm squared of a moment map

I am studying the paper by E. Lerman:
https://arxiv.org/abs/math/0410568
Let $(M,\sigma)$ be a connected symplectic manifold with a Hamiltonian action of a compact Lie group $G$, so that there exist ...

**2**

votes

**1**answer

96 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 ...

**1**

vote

**0**answers

38 views

### Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle

I am trying to understand the construction of symplectic inflation and I am stuck in the following point.
Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \...

**3**

votes

**0**answers

136 views

### Legendrian surgery and invertible elements in zeroth degree symplectic cohomology

Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...

**2**

votes

**0**answers

56 views

### What is the relation between the different generating functions thought as finite approximations of action functionals

In the book Introduction to symplectic topology by MC Duff and Salamon, a discrete analogue of the action functional is defined on $\mathbb{R}^{2n}$. The idea is that a Hamiltonian isotopy can be ...

**3**

votes

**0**answers

104 views

### Mixed characteristic in symplectic geometry

Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry?
There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...

**5**

votes

**1**answer

246 views

### Relationship between Gromov-Witten and Taubes' Gromov invariant

Fix a compact, symplectic four-manifold ($X$, $\omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic ...

**13**

votes

**2**answers

742 views

### Does Fukaya see all symplectic topology?

I recently had a debate with my friend about how much of symplectic topology is about Fukaya category. I thought that for the most part, symplectic topology is not about Fukaya category. Now, to prove ...

**6**

votes

**0**answers

91 views

### Maslov index of pair of paths in $\mathcal{L}(2n)$ and its relation with the Maslov index of a loop in $\mathcal{L}(2n)$

I'm reading [RS] and I was wondering what kind of connection there is between the Maslov index for a pair of paths $\lambda_0,\lambda_1 \colon [a,b] \to \mathcal{L}(2n)$ as defined in [RS] and the ...

**5**

votes

**0**answers

87 views

### A clarification in the definition of Seidel's absolute Maslov index for a pair of transverse Lagrangians

I'm reading Seidel's paper Graded Lagrangian submanifolds where he introduces the absolute Maslov index of a pair of graded lagrangians as follows:
Let $\mathcal{L}(V,\beta)$ be the Lagrangian ...

**3**

votes

**2**answers

172 views

### Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

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\...