# Questions tagged [symplectic-topology]

The symplectic-topology tag has no usage guidance.

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### Cohomological indices VS cup-length, Lusternik-Schnirelmann category, betti sum in critical point set theory

Let $M$ be a (closed) manifold acted on by a compact Lie group $G$. For any characteristic class $\alpha \in H^*(BG)$, one can define a so-called cohomological index of $M$ as follows:
$$\text{ind}_{\...

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108 views

### Normal coordinates for isotropic submanifolds

Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ in a neighbourhood $U$ of $p$ such ...

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327 views

### Moduli space of curves

Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space ...

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107 views

### Complement of Donaldson divisors in dimension 4

Let $(X,\omega)$ be a symplectic 4-manifold such that $\omega$ has a rational cohomology class. I am interested in Donaldson divisors (surfaces) $D$ in $(X,\omega)$ whose complement is a 1-handle body....

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176 views

### Compact simply-connected homogeneous symplectic manifold

I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...

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952 views

### What is the current status of the Arnold conjecture?

Let $(M, \omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth ...

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140 views

### Where can I find good surveys on Symplectic and Contact geometry

Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ?
In the symplectic case : I am ...

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102 views

### Unobstructedness of nodal holomorphic curve in symplectic manifold

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...

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375 views

### DGLA controlling deformation of holomorphic curves

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...

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116 views

### space of $\omega$-tame almost complex structures and $\mathrm{Diff}(M)$

Let $(M,\omega)$ be a symplectic manifold, and $J$ is an almost complex structure on $M$. $J$ is said to be $\omega$-tame if
$$
\omega(v, Jv) >0
$$
for all non-zero $v\in TM$.
It is commonly said ...

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73 views

### On the measure of regular and chaotic regions in a phase space

Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...

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185 views

### Non-trivial examples of overtwisted contact structures

Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\...

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149 views

### Recovering topological invariants of symplectic manifold from the group of Hamiltonian diffeomorphisms?

There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \...

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319 views

### Question about Obstruction Bundle Gluing paper of Hutchings-Taubes

I'm trying to learn about Embedded Contact Homology. To understand the proof of $d^2=0$, I started by watching Hutchings' lectures on Obstruction Bundle Gluing on YouTube (1, 2, 3) and have now ...

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81 views

### Is there a simply connected contact manifold, “non-exactly” fillable, cappable, such that the whole thing is symplectically aspherical?

Is there an example of a simply connected contact manifold W, with a non exact symplectic filling $M_1$, (that is, $M_1$ is a symplectic manifold, with contact boundary $W$ and a Liouville vector ...

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106 views

### Lagrangian foliation for a holomorphic symplectic manifold

I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...

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167 views

### symplectic topology of (perturbed) KAM tori

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...

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124 views

### Existence of isotopy preserving the action

Let $\gamma_1$ and $\gamma_2$ be simple closed curves in $R^4.$ Let $\lambda= x_1 dy_1+ x_2dy_2.$ Suppose that $\int_{\gamma_1} \lambda= \int_{\gamma_2} \lambda.$
I am looking for a reference for ...

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244 views

### Symplectic reduction of 4-manifolds with circle actions

Let $(M,\omega)$ be a $4$-dimensional closed symplectic manifold. Assume there exists a Hamiltonian $S^1$-action on $M$, let $\mu:M \to \mathbb{R}^*$ be its moment map and let $M_{\text{red}}=\mu^{-1}(...

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52 views

### Symplectic displacement energy for several intersection points?

Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as
$$
e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap ...

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290 views

### Two embedded symplectic spheres with zero square in a symplectic $4$-manifold

I am aware that the following result is a classical one (by now). But I am not able to understand who proved it. What should be a proper reference to this statement?
Theorem. Let $M^4$ be a compact ...

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354 views

### Chart for Deligne-Mumford Compactification

Suppose $g\ge 0$ and $n\ge 0$ are integers. We have the space $\overline{\mathcal M}_{g,n}$ of stable curves of arithmetic genus $g$ with $n$ marked points. The topology on this space is the one ...

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316 views

### Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program

My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...

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394 views

### Describing a Lefschetz fibration whose fiber is plumbing of $T^*S^n$

I am studying an example of symplectic Lefschetz fibrations. As far as I know, given a Weinstein manifold $F$ and a collection $V_1,\ldots,V_k$ of exact framed Lagrangian spheres of $F$, there exists ...

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125 views

### Existence of $K$-invariant complex structure

Let $K$ be a compact connected real Lie group, let $M$ be a symplectic manifold with a symplectic left action of $K$, and let $ω$
be the symplectic structure of $M$.
Does there always exist a $K$-...

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469 views

### $(M,\omega)$ not symplectomorphic to $(M,-\omega)$

Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$.
In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...

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160 views

### The structure of Banach manifolds in symplectic geometry

Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...

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### Is every contractible open bounded domain in $\mathbb R^{2n}$ symplecomorphic to a star-shaped domain?

In Hofer & Zehnder's book "Symplectic Invariants and Hamiltonian Dynamics" (Page 99) they present an example of a star shaped domain (bounded, with smooth boundary) in the shape of a "Bordeaux ...

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122 views

### Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus

Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...

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98 views

### Embeddings of the configuration space into the phase space of integrable systems

As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of ...

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171 views

### How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?

When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...

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### The isotopy class of a Boothby-Wang contact structure

Let $p:\Sigma\to M$ be a non-trivial principal $S^1$-bundle over a closed orientable surface $M$. Let's call $V$ the vector field generating the $S^1$-action. It is known that there exists a Boothby-...

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### Can one choose a sufficiently generic path of a.c.s such that only “codimension 1” bubbling occurs?

Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...

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136 views

### what are advantages of considering $\{J_z\}$-holomorphic curves for a parametrized family $\{J_z\}$ of almost complex structure?

Reference: McDuff-Salamon's book J-holomorphic curves and Symplectic Topology, second edition.
In section 6.7, the book introduces the moduli space of $\{J_z\}$-holomorphic curves (see page 184 for ...

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466 views

### Categorical mapping class group action

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action
$$ MCG(\Sigma) \to Auteq(\mathcal{C})$$
by ...

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437 views

### Almost complex structures on $\mathbb CP^2$ that are not tamed

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...

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615 views

### Gluing symplectic manifolds

The condition that allows gluing of symplectic manifolds, is the existence of a fixed point free $S^1$ action on the boundary, such that the orbits of the $S^1$ action are tangent to the kernel of the ...

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101 views

### Reference request: Liouville integrability of a torus action of small dimension on a symplectic manifold

Consider a hamiltonian toric acion on a connected real symplectic manifold of dimension 2n. The dimension of the torus, which we denote by $k$, may be less than $n$. The generators of the action will ...

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339 views

### Generating Fukaya category vs split-generating Fukaya category

I just started learning about Fukaya categories and got slightly confused by the following question. It looks like the statement that a collection of objects generate Fukaya category is stronger than ...

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### the tangent space $T_J\mathcal J^k$ of the space of $\omega$-compatible almost complex structure

Let $(M,\omega)$ is a symplectic manifold, and you can assume it is compact if necessary.
Denote by $\mathcal J^k = \mathcal J^k(M, \omega)$ the set of all $\omega$-compatible almost complex ...

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### infinite-dimensional transversality theorem and its application on the universal moduli space of pseudo-holomorphic curves

We would like to discuss Proposition 3.2.1 in McDuff&Salamon's book "J-holomorphic Curves and Symplectic Topology(Second Endition)".
Let me first remind you some background. Let $\Sigma$ be a ...

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224 views

### Gromov compactness theorem for genus $g >0$ Riemann surfaces

In McDuff & Salamon's book "J-holomorphic Curves and Symplectic Topology", only the Gromov compactness for spheres has been proved, and I wonder whether or not this compactness result is still ...

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513 views

### Definition of “Lagrangian skeleton”

I'm looking for the precise definition of "Lagrangian skeleton", as I'm eventually going to give a talk on this topic. As I asked a professor in my university about references on Lagrangian skeleta, ...

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73 views

### Regular almost complex structures on symplectic toric manifolds

Under which assumptions the almost complex structure J defined on a symplectic toric manifold is Fredholm regular for every J-holomorphic sphere?

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140 views

### Moser's argument for loops

Let $(\omega_t)_{t\in [0,1]}$ be a path of cohomologous symplectic forms on $X$. The standard Moser's argument shows that there exists a family of diffeomorphisms $(\psi_t)_{t\in [0,1]}$ of $X$ with $\...

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### Connectedness of the space of symplectic embeddings into a higher dimensional manifold

Suppose $M$ and $N$ are symplectic manifolds, $N$ is compact, $\dim_{\mathbb R}(N) \leq \dim_{\mathbb R}(M) -4$. Suppose there are embeddings $f_i:N \to M$, $i=0,1$ such that $f_i^*\omega_M$ is non-...

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168 views

### Hofer-Zehnder capacity of toric varieties

Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset \...

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184 views

### Grading in Lagrangian Floer homology

What are the conditions on a symplectic manifold (M,w) and on a Lagrangian submanifold L so that Lagrangian Floer complex CF(L, f(L)) is Z-graded? Here f is a compactly supported Hamiltonian isotopy. ...

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274 views

### How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...

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235 views

### Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact K\...