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Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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Moduli space of curves away from singular subsets

Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities. More specifically, I'm interested in ...
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Is there a canonical smooth structure on tame Fréchet orbit type stratifications?

In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...
MyShepherd's user avatar
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Expositions of symplectic reflection groups

We will work over $\mathbb{C}$. Remember that a finite subgroup $G$ of $\operatorname{GL}_n(\mathbb{C})$ is called a complex reflection group if it is generated by complex reflections $r$, which are ...
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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^*(...
onefishtwofish's user avatar
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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 ...
ccriscitiello's user avatar
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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)} \...
Nicolas Medina Sanchez's user avatar
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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 ...
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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 ...
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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?
warzasch's user avatar
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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 ...
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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 ...
CharlieHo's user avatar
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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 $(\...
user14334's user avatar
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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 ...
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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 ...
mathamphetamine's user avatar
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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 ...
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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 ...
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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 ...
arpa's user avatar
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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)\...
Someone's user avatar
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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 ...
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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 \...
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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,\...
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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!
kvicente's user avatar
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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 ...
jg1896's user avatar
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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 ...
Random's user avatar
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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 ...
jg1896's user avatar
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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 ...
jj_p's user avatar
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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-...
jg1896's user avatar
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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 (...
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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 ...
Someone's user avatar
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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....
warzasch's user avatar
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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 ...
Someone's user avatar
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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 ...
kvicente's user avatar
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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 ...
Someone's user avatar
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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 $M$ ...
Someone's user avatar
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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 ...
Uncool's user avatar
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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 ...
Ali Taghavi's user avatar
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1 answer
160 views

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\...
Adterram's user avatar
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3 votes
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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 $\...
Adterram's user avatar
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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. ...
Shuo Zhang's user avatar
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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 ...
Adterram's user avatar
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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}/\...
kvicente's user avatar
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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 ...
ChoMedit's user avatar
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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 ...
Riccardo's user avatar
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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 ...
sz3's user avatar
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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 ...
Alex Zukovich's user avatar
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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 ...
Ali Taghavi's user avatar
2 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 ...
Ali Taghavi's user avatar
3 votes
0 answers
75 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 ...
Giovanni Moreno's user avatar
2 votes
1 answer
65 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 ...
Someone's user avatar
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2 votes
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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 ...
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