# Questions tagged [symplectic-topology]

The symplectic-topology tag has no usage guidance.

249
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### When lagrangian fibrations are equivalent?

Given a $2n$-dimensional symplectic manifold $\mathcal{M}$ and two different lagrangian fibrations $\pi_1:\mathcal{M}\rightarrow \Gamma_1$ and $\pi_2:\mathcal{M}\rightarrow \Gamma_2$, with $\Gamma_1, \...

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### Liouville-Arnold and fibration relative to a convex polytope

Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...

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### Is the volume of the image of the moment map an invariant of a symplectic toric manifold?

Given a symplectic toric manifold is known that the image of the moment map is a Delzant polytope that fully characterises the manifold. What about the volume of the polytope? Is it possible that two ...

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### Symplectic form on the space of geodesic currents on a surface?

There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...

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### Gluing coordinate vector fields on a symplectic manifold [closed]

I wonder if there exists an open covering of a symplectic manifold for which the hypotheses for gluing coordinate vector fields are satisfied. In general, I don't think that coordinate vector fields ...

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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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### 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)\...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $M$ ...

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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 ...

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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\...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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 $(\mathbb{CP}^n,\omega)$ ...

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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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### 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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### 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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### Non-twist maps of the annulus and their lack of fixed points

I have been wondering about the existence of a kind of `counterexample' to a modification of the Poincare-Birkhoff theorem which badly breaks the twist condition.
Let me state a variant of the ...

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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 $...

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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, ...

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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 ...

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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?

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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 ...

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### Realizing closed manifolds as Legendrian submanifolds of the standard contact vector space

I started learning some basic contact geometry, in particular its flexible side, and I got stuck with the following naive question. Given a closed manifold of dimension $n$, we can always embed it ...

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### Choice of a family of almost complex structures when defining Floer Homology

Consider a $1$-periodic Hamiltonian $H:S^{1}\times M\rightarrow \mathbb{R}$ defined on a compact symplectic manifold $M$. Let's suppose $M$ is nice enough so that we can develop Floer theory on it. ...

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### Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"

I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$.
...

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### On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...

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### Gluing maps in Floer Homology and boundary conditions

Recently, I have been trying to a construction of a gluing map regarding the Lagrangian Floer Homology of two fibers in the cotangent bundle $T^*M$ of a manifold , in order to prove that the map $\...

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### Linearization of the Floer equation

In Floer Homology we want to prove that the Moduli spaces $\mathcal{M}(x^{-},x^{+})$ are finite dimensional manifolds. This is done by expressing them as the zero set of a Fredholm map. First one ...

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### Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?

Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...

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### Gluing of Morse-type trajectories in "Floer Homology of Cotangent bundles"

In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...

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### Gluing of hybrid trajectories in Floer homology

In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...

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### Banach manifold structure on the moduli space of hybrid trajectories

I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...

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### Multisymplectic connections and topological invariants

I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry.
From symplectic connections, it is well ...

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### Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove ...