Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,434
questions
3
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Integral equivariant formality for Hamiltonian T-actions
What is the simplest example of a compact symplectic manifold $M$ with Hamiltonian $T$-action for which the integral $T$-equivariant cohomology is not formal
i.e. $$H_T^*(M,\mathbb{Z}) \not \cong H^*(...
- 459
2
votes
0
answers
40
views
Explicit formula for complex structure on flag manifold/isospectral matrices?
Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
- 539
0
votes
2
answers
208
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Does this distribution exist?
Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
2
votes
0
answers
42
views
Definition of the vertical tangent bundle for a Lefschetz fibration
I realise I don't have a good definition for the vertical tangent bundle in the case of a Lefschetz fibration $(E,F)\to (S,\partial S)$ ($F$ is the subbundle given by the tangent bundle of the ...
- 2,018
1
vote
0
answers
121
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Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form
Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...
- 181
3
votes
0
answers
67
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Fundamental group of $\operatorname{Ham}(M,\omega)$ and higher homotopy groups
Given a symplectic manifold $(M,\omega)$, what do we know about $\pi_1(\operatorname{Ham}(M,\omega))$? What about higher homotopy groups?
- 179
2
votes
0
answers
121
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Is curvature of the canonical line bundle always $(1,1)?$
Let $(M,g,\omega)$ be a symplectic manifold with $g$ and $\omega$ denoting the Riemannian metric and the symplectic form respectively. If $J$ is a compatible almost-complex structure, then is the ...
- 853
4
votes
1
answer
100
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Fredholm property of linearization of Floer map
I am reading Audin and Damian's book "Morse theory and Floer homology". In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the ...
- 51
6
votes
1
answer
143
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Symplectic diffeomorphism of the cylinder moving a point to 0
I am currently reading though part of Zehnder's Lectures on Dynamical Systems. In Chapter VII, I have found myself in the following situation:
$Z(1)$ is a subset of standard symplectic space $(\...
- 133
7
votes
1
answer
426
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Goldman symplectic form vs Weil–Petersson symplectic form
I'm confused about the exact multiplicative factor that relates Goldman symplectic form on the $\operatorname{SL}(2,\mathbb R)$-character variety and the Weil–Petersson symplectic form on Teichmüller ...
- 38
2
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0
answers
136
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Spectrum of an almost Hamiltonian matrix
I have a complex-valued block matrix $N=\begin{bmatrix}
A & B \\
C & -A^*
\end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian.
If $C$ were Hermitian, $N$ would ...
- 116
1
vote
1
answer
181
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Special Darboux chart for tranverse Lagrangians
In the notes of Fukaya-Oh-Ohta-Ono (Lagrangian intersection Floer theory), Chapter 10 §54.1, it is stated:
Let $L_{1}$ and $L_{2}$ be a pair of oriented Lagrangian submanifolds in $(M, \omega)$ that ...
- 11
2
votes
1
answer
114
views
Symplectic compatification of a cotangent bundle, or of a neighbourhood of its zero section
Take a closed manifold $\mathcal{L}$ and endow its cotangent bundle $T^*\mathcal{L}$ by the standard symplectic form $\omega = -d\lambda$, $\lambda$ being the Liouville form.
I was wondering if it was ...
- 23
4
votes
0
answers
75
views
Pre-Symplectic Mapping
I have been studying symplectic integrators and their pre-symplectic extensions for dissipative systems. According to França, Jordan, and Vidal - On dissipative symplectic integration with ...
- 41
1
vote
0
answers
39
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Displaceability questions in the symplectic 2-sphere for level sets of a Morse function
Consider the symplectic $2$-sphere $S^2$ with the canonical symplectic form $\omega$. A subset $A$ is called displaceable if there exists $H:S^2\rightarrow\mathbb{R}$ smooth such that $\Phi_H^{1}(A)\...
- 781
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1
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225
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How to express the Euler-Lagrange equation in arbitrary coordinates where $\omega \neq \sum dp_i \wedge dq_i$
I posted my questions in a previous post MO, but it seems that a more refined version for question on the Euler-Lagrange equation is needed. So, I post my question again.
In standard symplectic ...
- 2,749
0
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1
answer
105
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How does the symplectic form $\omega$ manifests itself in the Euler-Lagrange equation? + Extreme confusion with time
Let $\omega$ be a symplectic manifold on $\mathbb{R}^n$ and the smooth function $H : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a Hamiltonian. For $p,q \in \mathbb{R}^n$ let us assume that
\...
- 2,749
1
vote
0
answers
67
views
Hyperkähler quotient of left $\operatorname{SU}(2)$-action on $(\mathbb{C}^2)^m \cong \mathbb{H}^m$
The natural $\operatorname{SU}(2)$-action on quaternions $\mathbb{H}\cong\mathbb{C}^2$ is hyperkähler. Extending it naturally to $\mathbb{H}^m$, one can make a hyperkähler quotient (where $\mu=(\mu_I,\...
- 1,627
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votes
1
answer
164
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Density of the set of convex polygons in the Banach-Mazur distance
Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!
- 181
4
votes
0
answers
125
views
Coulomb branches which are not of cotangent type
To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli ...
- 2,683
1
vote
1
answer
88
views
Derivative of the symplectomorphism evaluated at a point of the zero section of the cotangent bundle
It might be an easy question, possibly not worth posting here. In the proof of the Lagrangian neighborhood theorem, the authors have written the expression for the derivative of a symplectomorphism at ...
- 1,087
2
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0
answers
79
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Can Coulomb branches have symplectic resolutions?
My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb
branches ...
- 2,683
0
votes
0
answers
57
views
existence of moment maps for non-nef toric varieties
The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
- 533
6
votes
1
answer
234
views
Question about a remark on quantization of Coulomb branches
I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-...
- 2,683
6
votes
0
answers
195
views
Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
- 2,018
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votes
0
answers
21
views
The effect that applying a nodal slide has on the fibers above the eigenline
So recently I've come across the following question posed to me by myself. Suppose I have an almost toric fibration that was obtained from a Delzant polyope by applying a nodal trade. Now in this ...
- 781
3
votes
1
answer
308
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In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
- 179
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0
answers
38
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Potential function in the smoothing of toric degenerations when not collapsing all $-2$-Spheres
In the paper "An-type singularity and nondisplaceable Lagrangian tori", https://arxiv.org/pdf/1710.11221.pdf, by Sun the author proves that when taking the toric degeneration of a semi-Fano ...
- 781
1
vote
0
answers
115
views
Smooth action on cotangent space of the plane
Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by
$$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$
acts via ...
- 181
1
vote
1
answer
218
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Question about the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces"
In the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces" by Renato Vianna, the author constructs an infinite amount of non-symplectomorphic monotone Lagrangian tori in ...
- 781
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0
answers
61
views
Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians
In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ ...
- 781
2
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0
answers
131
views
What is the topology on the space of differential forms $\Omega^2(M)$?
I have posted this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here.
I have difficulty in understanding the meaning of "A ...
- 181
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172
views
Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
- 312
2
votes
1
answer
158
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Is a simple J-holomorphic curve injective everywhere except for finitely many points?
Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set
$$
Z:=\{z\...
- 1,361
3
votes
1
answer
288
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Image of the moment map
Let $(M^{2n}, \omega ,\mathbb T)$ be non-compact symplectic manifold with an effective and Hamiltonian torus $\mathbb T$-action.
Suppose its moment map $\mu$ is proper and the fixed point set of $\...
- 1,361
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votes
0
answers
158
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Structure maps of $\mathcal{A}_\infty$-bimodules
For Fukaya categories there are functors naturally induced by symplectomorphisms. Twisted versions of symplectic homology (fixed point Floer homology), open-closed maps and bimodules can be defined. ...
3
votes
0
answers
54
views
Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
- 1,361
3
votes
1
answer
410
views
Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
- 181
2
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answers
67
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Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
- 265
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86
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A couple of questions about the moduli space of annuli with some marked points on the boundary components
I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
- 2,018
3
votes
0
answers
62
views
Smooth Hamiltonian diffeomorphisms form a Baire space
Let $S$ be a closed surface equipped with an area form $\omega$. In Corollary 1.2 of this paper, Asaoka and Irie demonstrated that Hamiltonian diffeomorphisms which have a dense set of periodic points ...
- 219
0
votes
0
answers
79
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Symplectomorphism and Hamiltonian isotopy
I would like to ask whether a symplectomorphism of a given symplectic manifold respects Hamiltonian isotopy classes of Lagrangian submanifolds. In other words, given two Hamiltonian isotopic ...
1
vote
0
answers
143
views
Invariants associated to a principal bundle whose total space is a symplectic manifold acted symplectically by group structure
The following question - proposal came to my mind about 4 years ago but I did not find any solution to this question and did not find any answer via e-personal comunication with some ...
- 312
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votes
0
answers
249
views
When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$.
The pull back map $f^*$ is a symplectomorphism wrt the ...
- 312
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votes
0
answers
71
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
- 2,517
2
votes
1
answer
56
views
Displaceability of the sublevels below the Mane critical value
Recently I have been reading the paper "Symplectic topology of Mané's critical values" by Cielibak, Frauenfelder and Paternain. I am mostly interested in the part of the paper regarding the ...
- 781
2
votes
0
answers
60
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Confusion about proof of $C^0$ bounds for Floer curves on cotangent bundles
I have trouble understanding the proof of theorem 5.4 from Cielibak's article "Pseudo-holomorphic curves and periodic orbits on cotangent bundles". At the bottom of page 267 he defines a ...
- 21
2
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0
answers
31
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Kähler quotients for generic $\xi\in \mathfrak{g}^*$
In this question I intentionally omit words like "(non)compact" because I am not sure about the precise setting where this question makes sense.
Let $M$ be a symplectic manifold, $G$ a Lie ...
- 1,468
5
votes
0
answers
158
views
Is the wrapped Fukaya category a symplectomorphism invariant?
Say, let $\phi\colon W_1\to W_2$ be a symplectomorphism of Weinstein manifolds(or with stronger assumption that $W_1$ is Liouville homotopic equivalent to $W_2$, but with non-compact support), do they ...
- 271
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votes
0
answers
194
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Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?
Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories
...
- 131