# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,386
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### Reference for Morse-Bott vector fields

I'm looking for a reference for the following result:
Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ ...

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### Product structures in Rabinowitz Floer homology

Let $(M,d\lambda)$ be a compact exact symplectic manifold and $\overline{M}$ its symplectic completion. For simplicity we can think of $\overline{M}$ has a cotangent bundle and $\partial M$ the sphere ...

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### Topological obstruction for making a submanifold Lagrangian

Let $M$ be a manifold of symplectic type (we do not fix a particular symplectic form) and $N \subset M$ a submanifold of half the dimension. Is there a topological obstruction on the pair $(M, N)$ for ...

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### Examples of non-equivariant momentum maps

What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group ...

1
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1
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66
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### isotropy of the cotangent lift of a group action

I asked this question in stack exchange but have not received an answer, so I am posting it here.
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the ...

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### Deformation quantization of an integrable system

What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?

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### Bubbling off a sphere in a splitting/stretching manifold

This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology ...

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### Index of Floer operator for Hamiltonian vs Lagrangian Floer Homology

I am trying to see if there is a way to translate the computation of the index of the Floer operator for Hamiltonian Floer to Lagrangian Floer. Hamiltonian Floer homology is a theory that counts (...

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1
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### Obstructions to maximal number of independent constants of motion in a given symplectic manifold

Given a compact symplectic manifold $(X, \omega)$, are there any invariants (topological or easily computable geometric/analytic ones) which give an estimate of the maximal number of independent ...

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### Simple examples of Gromov-Witten invariants not being enumerative

I understand why Gromov-Witten invariants in general are not enumerative, so it's not necessary to explain this. However to test something I'm working on, I'm looking for examples of concrete ...

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### Seidel's calculation of the Floer cohomology of a cotangent fibre and its Dehn twist

I am reading Seidel's paper on exact Lagrangian submanifolds in $T^*S^n$ and the graded Kronecker quiver, and in Lemma 2 (2) he claims the following fact: if $F_0$ is a cotangent fibre and $F_1$ is $\...

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### Hamiltonian action as a group homomorphism

It is sometimes demanded that a Hamiltonian group action $G \times M \to M$ allow for a Lie algebra homomorphism from $\mathfrak{g}$ to $C^\infty(M)$ with the Poisson bracket.
Is there a natural ...

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### Connected components of Isotropy types as strata of Poisson leaves

Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$.
We can say that $X$ is trivially a normal variety ...

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### Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials

I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...

4
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1
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### In what topology does Gromov's lemma hold on noncompact symplectic manifolds?

In symplectic geometry, it is commonly said that ``the set of almost complex
structures tamed to a symplectic form is contractible'' even on noncompact symplectic manifolds. In my understanding
one ...

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### Symplectic quasi-states and displaceability of subsets of symplectic manifolds

In the paper "Quasi-states and symplectic intersections", Entov and Polterovich, introduced the notion of a partial symplectic quasi-state and used it the prove the following theorem:
Let $(...

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### How to calculate the exterior derivative on manifolds of smooth mappings?

Let $S$ be a compact finite-dimensional manifold $S$ and $(M, \omega)$ a symplectic manifold. The space of smooth maps from $S$ to $M$, denoted by $\mathcal{M}$, has a canonical infinite-dimensional ...

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### Explicit Lagrangian fibrations of a K3 surface

I would like to look at the behaviour of the fibres of a Lagrangian fibration (such that at least some fibres are not special Lagrangian) $X\to\mathbb{CP}^1$ under the mean curvature flow (in relation ...

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### Why are symplectic toric varieties projective?

Let $X$ be a symplectic toric manifold meaning a compact symplectic manifold $(X, \omega)$ with $\dim{X} = 2n$ equipped with a Hamiltonian action of a maximal-dimension torus $\mathbb{T} = (\mathbb{S}^...

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### Weinstein's neighborhood theorem for exact symplectic manifolds

Let $(M, \omega)$ be a symplectic manifold and $L\subset M$ a Lagrangian submanifold. The Lagrangian Neighborhood Theorem says that there exists a neighborhood $U$ of $L$ in $M$ and a ...

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### Seeing $\mathbb{CP}^2 \mathbin\# \overline{\mathbb{CP}^2}$ as a symplectic reduction of different manifolds

I have been reading the paper "Remarks on Lagrangian intersections on toric manifolds" by Abreu and Macarini, which gives several non-displaceability results by avoiding the use of ...

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### What are known properties of the boundary curves of J-holomorphic curve with boundary

Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...

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1
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69
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### Deformation of a Liouville form with a diffeotopy

Let $M$ be a surface with boundary and let $f_t: M \to M, t \in [0,1]$ be a differentiable family of diffeomorphisms (I think this is usually called a diffeotopy). Suppose I have a Liouville form $\...

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### A geometric interpretation of the fractional Fourier transform

I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...

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### Lagrangian cobordisms from a Legendrian knot to its scaled version

Having a Legendrian knot L in $\mathbb R^3$ and its scale aL (the length of Reeb chords of it are scaled by a>0), are these two Legendrians Legendrian isotopic? Maybe weaker, is there an exact ...

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114
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### Transformation of Morse function in $\mathbb{CP}^n$ trough symplectomorphism

I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.
Consider $(\mathbb{CP}^n,\omega)$ ...

4
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135
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### The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...

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### Is there a relation between symplectic toric orbifolds and semi-toric systems?

So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ ...

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### Understanding the Lagrangian Lift of a given isotopy

I was looking at the paper "Sheaf quantization of Hamiltonian isotopies
and applications to non-displaceability
problems" by Guillermou-Kashiwara-Schapira(GKS). So firstly, GKS doesn't ...

6
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1
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277
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### Homology and cohomology of free loop spaces

String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$.
Let $k$ be a field and let $M$ be $n$-...

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### Symplectic embedding of the ellipsoid $E(1, 4)$ into the ball $B(2)$

Let $\omega_0=dx_1 \wedge dy_1 + dx_2 \wedge dy_2$ be the standard symplectic form on $\mathbb{R}^4$, $E(a,b)=\{(x_1, y_1, x_2, y_2)\in \mathbb{R}^4: \frac{\pi(x_1^2 +y_1^2)}{a}+\frac{\pi(x_2^2 +y_2^2)...

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78
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### Question about symmetric bilinear form and convex geometry

Consider a Finsler norm $\varphi$ in $R^4$, and a positive definite symmetric bilinear form $S$, there also exists a vector-valued symmetric linear form $Q$, such that if $S(v, w) = 0$, then $Q(v, w)$ ...

6
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### Progress on composition of Lagrangian correspondences/definition of symplectic categories?

I am interested in Lagrangian correspondences in the context of symplectic manifolds, namely Lagrangian submanifolds $L_{12}$ of $M_1\times \bar M_2$ where $M_1$ and $M_2$ are symplectic manifolds ...

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### Question on the proof of doing a nodal trade, almost-toric fibrations

I am trying to understand the details of the proof of lemma $6.3$ of the following notes https://arxiv.org/pdf/math/0210033.pdf, which give us specific conditions of when we can swap a neighborhood of ...

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### Doing a nodal trade in a semi-toric system

Recently I have been studying semi-toric systems and almost toric fibrations. For the purpose of semi-toric fibrations I have been reading these notes https://arxiv.org/pdf/math/0210033.pdf. ...

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### Could this be a Hamiltonian symplectomorphism on a symplectic toric manifold

Suppose we have a sympletic toric manifold $(M,\omega)$ of dimension $4$ and let $\triangle$ be its corresponding Delzant polytope. Suppose that this polytope is "nice" enough so that we are ...

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133
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### Neighborhood theorem for conical Lagrangian

Let $(M,\omega)$ be a compact $2n$ dimensional symplectic manifold and $T$ be a compact smooth $(n-1)$ dimensional manifold.
Let $CT$ be the cone over $T$, i.e. $CT = T\times [0,1] / \sim $ where $\...

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### Generators of the symplectic group

Let $n$ be an integer $\ge 1$. We define the $2n\times 2n$ matrix $\sigma$
with $n\times n$ blocks by
$$
\sigma=\begin{pmatrix}0&I_n
\\-I_n&0\end{pmatrix}.
$$
The symplectic group $Sp(n)$ is ...

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0
answers

102
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### Doubt in the proof of Mcduff''s method of probes

I have been reading the paper "Displacing Lagrangian toric fibers by probes" by Dusa Mcduff, here is the arxiv link https://arxiv.org/pdf/0904.1686.pdf.
I have a doubt in the proof of lemma $...

4
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98
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### Overtwisted contact structures on $S^3$

All the isotopy classes of overtwisted contact structures are classified by the Hopf invariant. Are any of these contact structures contactomorphic?
Suppose $d_{3}(\xi_{n}) = n$, then my guess is that ...

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### Displaceability questions of fibers on integrable hamiltonian systems

Alot is known about the (non)-displaceability of the fibers of a toric symplectic manifold. For example there is Mcduff's method of probes to prove displaceability results using the moment polytope, ...

0
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1
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133
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### Question about coadjoint orbits of compact connected Lie groups

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Denote by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $r$ be an element of $\mathfrak{g}^*$ such that $G_r$ the stabilizer of ...

2
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### Question on Gromov-Witten invariants when $A=0$

I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's ...

3
votes

1
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### Is there an analogy of Austin-Braam approach to Bott-Morse type Hamiltonian Floer homology?

Austin-Braam approach uses the multicomplexes of de Rham complex on critical submanifolds to describe Bott-Morse theory.
For more details, see the follows:
https://link.springer.com/chapter/10.1007/...

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0
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### Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions

Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...

2
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70
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### Weinstein fillings of a unit cotangent bundle

Given a closed, orientable manifold M, and its unit cotangent bundle $ST^{\ast}M$. I wonder under which conditions $ST^{\ast}M$ admits a subcritical Weinstein filling?

4
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### Almost toric mutations

I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,...

3
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### Smooth handle attachment vs Weinstein handle attachment

Given a closed smooth manifold $M$ of dimension $n$, to which we attach a $k$-handle $H_k$.
Take $T^{\ast} M$, can one realize $T^{\ast} (M\cup H_k)$ as a result of symplectic or Weinstein handle ...

6
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1
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442
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### Are holomorphic Lagrangians locally graphs?

Let $(M, \omega)$ be a holomorphic symplectic manifold of (complex) dimension $2n$. Let $x$ be a point in $M$. My understanding from the discussion and answers to this MO question is that there exists ...

4
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### Isotopy classes of $CP^1$ in 4-manifolds

Let $S_1$, $S_2$ be homologous embedded 2-spheres
in a compact smooth 4-manifold. Under which additional
conditions are they smoothly isotopic? I am interested
in the state of the art picture when $...