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1answer
123 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 ...
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3answers
890 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
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1answer
132 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
312 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
142 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 ...
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3answers
346 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
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2answers
418 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
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1answer
166 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 ...
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2answers
381 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 $X=\{(p,f(p)): p\in M\}$, ...
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1answer
234 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
455 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
326 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 = ...
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3answers
287 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
431 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
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2answers
658 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 ...
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1answer
216 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
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1answer
292 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
532 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?
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0answers
405 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 ...
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3answers
767 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
429 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?
2
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2answers
463 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
404 views

Almost-Kahler Einstein four manifolds

Are the odd Betti numbers of an Almost-Kahler Einstein four manifolds necessarily even ?
4
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1answer
353 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 ...
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4answers
744 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 ...
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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
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1answer
412 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
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1answer
440 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 ...
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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
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1answer
251 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
827 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
512 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
277 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 ...
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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
712 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 $\pi_1$ ...
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4answers
3k views

Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general. What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...