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0
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0answers
40 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 ...
10
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6answers
498 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, ...
2
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0answers
125 views

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 ...
1
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0answers
63 views

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 ...
1
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0answers
59 views

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 ...
3
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0answers
78 views

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 ...
3
votes
2answers
158 views

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 ...
4
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0answers
100 views

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 ...
2
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1answer
128 views

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 ...
7
votes
2answers
241 views

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 ...
4
votes
0answers
118 views

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 = ...
2
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0answers
165 views

$C^0$ 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 ...
0
votes
1answer
195 views

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 ...
1
vote
1answer
148 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
1
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1answer
162 views

Question about transversality for PSS map in Hamiltonian Floer cohomology

Let X be a compact symplectic manifold and $H_t,J_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a ...
12
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3answers
1k views

Is there a physical intuition for Darboux's theorem?

We know that there is a physical interpretation for symplectic manifolds (briefly, the fact that a sympletic form assigns to any Hamiltoninan a vector field which describes the motion of particles). ...
5
votes
1answer
186 views

Does the smooth manifold $\#_{l}CP^{2}\#_{k}(-CP^{2})$ admit a symplectic structure?

Let $-CP^{2}$ denote the complex projective surface $CP^{2}$ with the reverse orientation. I have seen some results about the existence of symplectic structures on the connected sums ...
8
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1answer
357 views

Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a compatible almost complex structure $J$, such that the symplectic form determines an integer cohomology class, ie $$ [\omega] \in H^2(M, ...
3
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1answer
163 views

Can symplectic blow up increase symplectic capacities?

Let $N$ be a symplectic submanifold of $M$. Symplectic blow up of $M$ along $N$ is an operation replacing a tubular neighborhood of $N$ with the projectivization of that neighborhood. So it decreases ...
3
votes
3answers
428 views

Symplectic blow-up

Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the ...
7
votes
2answers
476 views

How to prove Arnold Conjecture without using S^1 localization?

By the Arnold Conjecture, I mean the following statement: Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# ...
1
vote
1answer
194 views

A basic question related to Hamiltonian isotopy in symplectic geometry

In any standard symplectic geometry/topology textbook, the concept of Hamiltonian isotopy was introduced: $(M, \omega)$ is a sympplectic manifold. Given a symplectic isotopy $\phi_t : M \rightarrow ...
1
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2answers
393 views

On the de Rham cohomology of 1-forms in cotangent bundle.

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a natural Lagrangian submanifold of it. A well known result is that any submanifold $X=\{(p,f(p)): p\in M\}$, ...
0
votes
1answer
235 views

$q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define $\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$ by $\perp(\omega)=i_{X_{\Omega}}(\omega)$ here if ...
6
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2answers
475 views

A Question on Exterior Forms

For the last few days, I have been trying to answer the following algebraic question in exterior algebra. The following question appears as an algebraic step in the context of existence of solutions ...
6
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2answers
339 views

Square root for Hamiltonian diffeomorphisms

Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then $$ \psi_1 = ...
3
votes
3answers
304 views

Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...
8
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0answers
460 views

Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as ...
7
votes
2answers
685 views

Relative version of Symplectic Thom conjecture.

Ozsváth and Szabó proved Symplectic Thom conjecture [Annals of Mathematics, 151(2000), 93-124]. It states: An embedded symplectic surface in a closed, symplectic 4-manifold is ...
1
vote
1answer
220 views

codimension of bubbling of disk and sphere

I would like to understand why bubbling of disks are said to be co-dimension 1 phenomena and bubbling of spheres co-dimension 2 phenomena.
4
votes
1answer
298 views

“Rounding the corners” to get contact boundary

Suppose we have symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ with non-empty boundary of contact . Often we need to deal with the product $M_1 \times M_2$ with the product symplectic ...
3
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1answer
561 views

Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6?

Are there any symplectic but not complex Calabi- Yau manifolds in real dimensions 4 and 6?
8
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0answers
429 views

SFT gluing on chain level in Floer homology?

I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and ...
12
votes
3answers
859 views

How transitive are the actions of symplectomorphism groups ?

This question is motivated by the classical fact from differential geometry : Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ ...
2
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1answer
442 views

The Hard Lefschetz property on Almost-Kahler manifolds

In the realm of almost-Kahler geometry , to what extent , the hard Lefschetz property is still holds?
3
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2answers
484 views

Reference for Almost-Kahler geometry

Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?
4
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0answers
415 views

Almost-Kahler Einstein four manifolds

Are the odd Betti numbers of an Almost-Kahler Einstein four manifolds necessarily even ?
4
votes
1answer
354 views

Is every contractible loop contained in a Darboux chart?

Let $(M,\omega)$ be a symplectic manifold and $\gamma:S^1 \rightarrow M$ be a contractible smooth loop. Is it possible to find an open set $U \subset M$ such that $\gamma(S^1) \subset U$ and such ...
0
votes
4answers
830 views

What's dual torus and mirror manifold?

I guess this is a well known fact/definition for many people. It is mentioned in many places that if $\Gamma$ is a lattice of a vector space(vector bundle/affine bundle) $V$, then there is a dual ...
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4answers
1k views

When are two symplectic forms “isotopic”?

I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long ...
8
votes
3answers
1k views

Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?

Let $G=\mathbb{Z}/2\mathbb{Z}$. Let $f\colon L \to N$ be a smooth map of connected smooth closed $n$-dimensional manifolds such that the induced map $f^\* \colon H^\*(N,G) \to H^\*(L,G)$ is an ...
7
votes
1answer
419 views

Real interpretations of Discontinuities in Floer homology

This question is motivated by the answer in this question (you dont have to read it to understand the following). I am not that proficient in calculating Floer homology, and I held back on answering ...
10
votes
1answer
454 views

Hamiltonian displaceability of tori in symplectic balls

Here is my first try at a question, which is a really easy to state question about displaceability: Let $D$ be the unit disk in the complex plane $D = \{ |z| \leq 1 \}$ equipped with its standard ...
9
votes
5answers
2k views

Understanding moment maps and lie brackets

I'm trying to learn about moment maps in symplectic topology (suppose our Lie group is G with lie algebra g, acting on the symplectic manifold (M,w) by symplectomorphisms). I'm having a hard time, and ...
2
votes
1answer
254 views

Algebraic geometric model for symplectic $T^* \Sigma_g$?

I was aware of an algebraic geometric model of symplectic $T^* S^2$ recently, that it is $\{x_1^2+x_2^2+x_3^2=1\}$ in $\mathbb{C}^3$, which the Lagrangian $S^2$ is just the real part, and in this way ...
12
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2answers
907 views

Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I found this in claim a ...
2
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2answers
530 views

Why (and whether) is any smooth embedded torus in R^4 isotopic to an embedded Lagrangian torus?

The question is pretty self-explanatory; we are dealing with the standard symplectic structure on ℝ4. Some background: I'm reading the thesis "Lagrangian Unknottedness of Tori in Certain ...
3
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1answer
287 views

Anosov diffeomorphisms and the chaotic hypothesis

There is a well-known "chaotic hypothesis" dating from 1995 or so in statistical physics that suggests that classical statistical-physical systems should be "effectively" Anosov. I won't get into the ...
11
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4answers
1k views

Periodic orbits of Hamiltonian systems

Given a Hamiltonian $H$ on $\mathbb{R}^{2n}$ and a periodic orbit $\gamma$, what in general can one say about the existence of periodic orbits near $\gamma$? I'm almost embarrassed to ask this ...
8
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4answers
727 views

How are invariants represented in category theory?

I'm trying to better understand how to think about invariance in the setting of category theory. In some cases it seems there's an obvious interpretation: for instance, the fundamental group $\pi_1$ ...