The symplectic-topology tag has no usage guidance.

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### 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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61 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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**1**answer

163 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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20 views

### Extending a symplectomorphism of a surface with boundary, properly embedded in a 4-disk, to the ambient space

Let $S$ be a surface with boundary that is properly embedded, via a map $f$
say, inside a $4$-disk. When is the embedding isotopic to the identity? --
All embeddings and isotopes are in the symplectic ...

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198 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\...

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71 views

### Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...

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73 views

### Vanishing first Chern class on fibers and Symplectic reduction

Suppose that a Lie group $G$ acts on compact Kahler manifolds $M$ and $N$ via symplectomorphisms take $\eta\in \mathfrak g^*$. and let $M_\eta$ and $N_\eta$ are symplectic quotients of $M$ and $N$ ...

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124 views

### Example where Calabi invariant is nontrivial?

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of $...

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101 views

### Triangulation of S^2xS^2

Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ?
Thanks,

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64 views

### Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants before....

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46 views

### Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...

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105 views

### holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...

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164 views

### Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...

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179 views

### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...

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**1**answer

179 views

### Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...

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141 views

### Maslov class as a relative cohomology class in H^2(M, L)

Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to ...

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**1**answer

259 views

### Steps in paper on sympl. geometry unclear

I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.
It deals with the question how the stability properties of a ...

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**1**answer

176 views

### Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a ...

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**2**answers

245 views

### Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...

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174 views

### Homology class of a Lagrangian Klein bottle

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...

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### Flux group of surfaces with genus $g\ge2$

I have read in this paper by C. Campos-Apanco and Pedroza (at the end of page 2) that:
When $(M,\omega)$ is a closed surface of genus greater than one, the flux group is trivial.
Does anyone ...

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**1**answer

544 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

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55 views

### Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...

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**1**answer

115 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

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77 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

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109 views

### Generation of compact Lagrangians over fields with characteristic 2

Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...

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### McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...

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### Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional $\...

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330 views

### Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 \omega(\dot{x}(t)-X_H,Y)...

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253 views

### On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...

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136 views

### Complex but not Symplectic

For every $n$ there exist a smooth manifold $M$ of $dim M = n$ that admits a complex structure but not a symplectic one?

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### why is there such a 1-form on a planar open book?

Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform page-...

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### Lagrangian fibrations with isolated singular fibers

Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold ...

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### Fully faithful embedding of the exact Fukaya category

Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...

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### Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$

I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...

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### Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...

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**2**answers

389 views

### When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?

My goal is to better understand the Maslov-index of pseudoholomorphic disks.
For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic ...

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108 views

### Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is:
Is the group of symplectic automorphisms
of $S^2$ with respect to this ...

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**6**answers

943 views

### When do you go hunting for Lagrangian submanifolds?

Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...

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### Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres

Given a symplectic 6-manifold $(M,\omega)$ and an embedded
symplectic 2-sphere $C\subset M$ whose normal bundle has the first
Chern class -2. How to find on $M$ another closed 2-form $\eta$ which
only ...

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### Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$
acts holomorphically such that the group $K$ preserves the
Kähler ...

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### Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...

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### The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative
h-principle cannot imply a positive solution to the problems for
extending symplectic structures.
The following is a relative h-...

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**2**answers

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### Reference Request: “Neck Stretching Procedure” (In Symplectic Field Theory)

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/...

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### Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...

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### On Lerman's description of symplectic cut

Assume $(X,\omega)$ is a compact real $2n$-dimensional symplectic manifold with a Hamiltonian torus action corresponding to the moment map $\mu:X\to \mathfrak{t}^*\cong \mathbb{R}^k$.
In this ...

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### The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...

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### What can be said about compact embedded exact Lagrangians in the generalized pair of pants?

What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...

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### $C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...

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### A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$
is integral, for some almost complex structure $...