Hamiltonian systems, symplectic flows, classical integrable systems

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1answer
95 views

Is the space of half-dimensional symplectic linear subspaces of $\mathbb{R}^{4n}$ which trivially intersect $\mathbb{R}^{2n}\times\{0\}$ contractible?

I would like to know whether the subspace of the symplectic Grassmanian $Gr_{2n}^{Sp}(\mathbb{R}^{4n})$ consisting of symplectic linear subspaces in $\mathbb{R}^{4n}$ which have dimension $2n$ and ...
5
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3answers
743 views

what prevents a manifold to be symplectic?

Are there any obstructions known which prevent an even dimensional orientable manifold from being symplectic? I am a novice in this area so I unfortunately I cannot make the question more precise. ...
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0answers
52 views

action aifference for cylinders in a symplectic cobordism

Suppose we have a contact manifold (M,$\xi$) and two associated contact forms $\alpha$ and $\beta$ s.t.$\beta=f\alpha$ with $f>0$. Suppose also that we have two almost complex structures $J_\alpha$ ...
4
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1answer
407 views

Recognizing the stablizer of a degenerate three forms in six dimension

Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }. we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0} Let $V\cong ...
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1answer
174 views

Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf. Is there some sense, or some ...
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2answers
257 views

Proof of Arnold Conjecture for monotone symplectic manifolds

I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf: Namely the author on page 32 ...
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1answer
149 views

Condition in proof of the Arnold conjecture for monotone manifolds

In http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf the Arnold conjecture from Symplectic Geometry is shown for the case of montone symplectic manifolds $(M, \omega)$ (i.e. we have $\int_{S^2} ...
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1answer
278 views

What foliations are symplectic foliations?

On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a ...
10
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1answer
839 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
5
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2answers
367 views

What information is required for SYZ mirror symmetry?

Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...
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1answer
419 views

why the group $GL(6,V)$ has an open orbit?

N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written ...
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1answer
471 views

Known size invariant for Riemannian manifolds?

Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under ...
3
votes
2answers
288 views

Uniqueness of Kähler form with same volume

Hallo, Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...
7
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2answers
458 views

Orientations for pseudoholomorphic curves with totally real boundary condition

I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition. I believe that Fukaya-Oh-Ohta-Ono have shown that if ...
5
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0answers
259 views

Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal ...
5
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1answer
276 views

Questions on how SYZ conjectures is deduced from HMS conjeture.

The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) ...
4
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1answer
211 views

Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...
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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 ...
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0answers
71 views

Preimage of regular values of a admissible hamiltonian

Hi everyone: I have a question about a proof in Siburg's paper "Symplectic Capacities in Two Dimensions". We define a admissible Hamiltonian $H$ in such a way that near from the boundary of a ...
2
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1answer
194 views

Hamiltonian actions and contractible loops

Let $(M, \omega)$ be a symplectic manifold and $G$ be a compact Lie group. Suppose we have a Hamiltonian $G$-action on $M$, with moment map $\mu: M \to {\mathfrak g}^*$. We assume that the moment map ...
4
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1answer
198 views

Rotations, Harmonic Oscillators, Gaussians, Ladders

I am trying to understand better the quantization of the Harmonic Oscillator. Here are three ways of thinking about the Harmonic Oscillator. Eigenfunctions of the differential operator: $H = ...
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2answers
530 views

Applications of Floer homology

Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture. Every answer would be appreciated.
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3answers
806 views

Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...
3
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1answer
251 views

Igusa invariants of genus 2 curves as Siegel modular functions?

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...
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1answer
241 views

Energy quantization for $J$-holomorphic spheres

Let $(\mathbb{CP}^1, j, g_{\text{FS}})$ be the complex projective line with the standard complex structure and the Fubini-Study metric and let $(M,J,\omega,g)$ be an almost Kähler compact manifold ...
4
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1answer
614 views

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that ...
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0answers
199 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
6
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1answer
190 views

Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section. Recall that an ...
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1answer
139 views

Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

Hallo, Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
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0answers
177 views

Different complexifications of a real analytic Riemannian manifold

Hi, I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...
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1answer
202 views

Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo, It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
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 ...
1
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2answers
274 views

Analytic Lagrangian Submanifolds

Hallo, I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
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2answers
724 views

Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times ...
0
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1answer
226 views

Polarisation in a nighbourhood of a Lagrangian submanifold

Hallo, Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such ...
1
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1answer
163 views

if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?

Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form $q: \Pi \oplus \Pi \rightarrow R $ $\big( \alpha ,\beta \big) \longrightarrow \alpha ...
1
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1answer
270 views

Cotangent space of the sphere

In analyzing the spherical pendulum the cotangent space of the sphere is defined as $ T^*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$ my problem with ...
1
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1answer
147 views

Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it. Here ...
5
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2answers
917 views

What is geometric intuition of special Lagrangian manifolds?

Let $M$ be (for example) a Calabi-Yau threefold with Kaehler form $\omega$ and holomorphic 3-form $\Omega$. We say that a submanifold $L$ of $M$ is a special Lagrangian submanifold if $L$ is ...
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0answers
118 views

Relation between Adpted Complex Structure and Hyperkaehler Structure

Hallo, I am reading the paper "Hyperkaehler structures on total spaces of holomorphic cotangent bundles" by Kaledin where he puts a hyperkähler structure on a neigbourhood of the $0$-section in the ...
3
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1answer
239 views

Spin-c Structures with Near-Symplectic Forms

Consider a smooth compact oriented 4-manifold $X$. Although not all 4-manifolds admit a spin structure, they do admit spin-c structures. And if $X$ does admit a spin structure, then there is a ...
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2answers
283 views

Delauney triangulation in high (>20) dimensions

Hi all, I know that its very hard to find the Delauney triangulation of high dimensional spaces, especially if there are several thousand points that need to be triangulated. So I was wondering . . ...
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4answers
1k views

How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...
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0answers
371 views

Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper. I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...
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1answer
267 views

Holonomy group of a non-compact Kaehler manifold

Hallo, I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form ...
3
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1answer
223 views

Decorations in Szabo's combinatorial spectral sequence

Szabo in http://arxiv.org/abs/1010.4252 gives a combinatorial candidate for what an explicit calculation of the spectral sequence of branched double covers should yield. In other words he gives a ...
6
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4answers
577 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by \begin{equation} ...
2
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1answer
550 views

Computing the cohomology class of a symplectic form

If a symplectic manifold $(M,\omega)$ is given, is there any efficient method to compute $[\omega]$, the cohomology class of the symplectic form? I do know that efficient is not a good word to ...
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5answers
689 views

Examples of non-Kahler compact symplectic manifolds.

I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it. Please avoid giving repetitive ...
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0answers
143 views

Level sets of momentum map for diagonal action on two coadjoint orbits.

Hi, I'm trying to get a better understanding of multiplicities in geometric quantization, and so I've been concentrating on a specific simple case: let $\mathcal{O}\subset\mathfrak{g}^*$ be an ...