Hamiltonian systems, symplectic flows, classical integrable systems

**5**

votes

**1**answer

269 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**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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 ...

**0**

votes

**0**answers

70 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**

votes

**1**answer

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**

votes

**1**answer

196 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 = ...

**3**

votes

**2**answers

502 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.

**7**

votes

**3**answers

777 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**

votes

**1**answer

246 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 ...

**12**

votes

**1**answer

239 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**

votes

**1**answer

601 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 ...

**7**

votes

**0**answers

193 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**

votes

**1**answer

188 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 ...

**1**

vote

**1**answer

138 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 ...

**6**

votes

**0**answers

171 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 ...

**1**

vote

**1**answer

194 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**

votes

**2**answers

473 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**

vote

**2**answers

272 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 ...

**13**

votes

**2**answers

715 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**

votes

**1**answer

224 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**

vote

**1**answer

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**

vote

**1**answer

251 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**

vote

**1**answer

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**

votes

**2**answers

887 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 ...

**0**

votes

**0**answers

117 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**

votes

**1**answer

234 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 ...

**1**

vote

**2**answers

260 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 . . ...

**11**

votes

**4**answers

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 ...

**4**

votes

**0**answers

363 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 ...

**1**

vote

**1**answer

266 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**

votes

**1**answer

217 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 ...

**5**

votes

**4**answers

568 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**

votes

**1**answer

537 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 ...

**1**

vote

**5**answers

666 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 ...

**1**

vote

**0**answers

139 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 ...

**4**

votes

**2**answers

412 views

### $J$-holomorphic curve as a minimal surface

The following is a part of the proof of Gromov nonsqueezing theorem.
The existence of a $J$-holomorphic curve gives an upper bound for the radius of a symplectically embedded ball.
Let $\psi: B(r) ...

**6**

votes

**1**answer

780 views

### 'Contactization' and Symplectization

Given any manifold $M$, we can get a symplectic manifold by taking the cotangent bundle $T^\ast M$ with symplectic form $\omega=\sum dp_i\wedge dq_i$. Given any manifold $M$, we can get a contact ...

**5**

votes

**1**answer

289 views

### The Fukaya category of a simple singularity (reference request)

I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...

**2**

votes

**2**answers

288 views

### What are the possible symplectic structures on a given Lie groupoid?

Recall the definition of a symplectic groupoid. Roughly this is a Lie groupoid such that the object manifold is Poisson, and the arrow manifold is symplectic such that the symplectic form is ...

**16**

votes

**3**answers

1k views

### Why are Gromov-Witten invariants of K3 surfaces trivial?

Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...

**3**

votes

**1**answer

270 views

### Sarkar's Maslov index formula

I have difficulty understanding Sarkar's maslov index formula in symmetric products from http://arxiv.org/abs/math/0609673.
If $D$ is an $n$-sided region with corner points $p_1,\ldots, p_n$ then it ...

**5**

votes

**2**answers

702 views

### Almost Complex Structures: 'Tame' versus 'Compatible'

Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures (a.c.s. for short). These are $J:TM\to TM$ with $J^2=-\text{id}$. An a.c.s. $J$ is $\omega$-tame when ...

**2**

votes

**1**answer

161 views

### Density in sc Banach spaces and polyfold theory

Part of the definition of a sc Banach space $E=E_0 \supset E_1 \supset \cdots$ is that $E_\infty = \bigcap_m E_m$ is dense in each $E_m$. Unfortunately, this rules out scales of Hölder spaces ...

**6**

votes

**2**answers

280 views

### Why is the base of SLAG fibration of CY3 expected to be $S^3$?

The SYZ conjecture roughly says that any Calabi-Yau threefold $X$ has a special Lagrnagian fibration $\pi:X\rightarrow B$ by 3-tous with section and one of its mirror partners $\check{X}$ is obtained ...

**5**

votes

**1**answer

411 views

### Complex structures on a K3 surface as a hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...

**0**

votes

**0**answers

203 views

### How can we formulate maximal time $T$ in Hyperbolic Kahler Ricci ﬂow

In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by ...

**2**

votes

**2**answers

424 views

### Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure?

Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$).
My questions is: it is always true that we have ...

**12**

votes

**1**answer

374 views

### What structure on the second order cotangent bundle ?

It is well-known that the total space of the cotangent bundle $T^*X$ of a given smooth manifold $X$ admits a symplectic form $\omega$. It is actually exact: $\omega=d\lambda$. The $1$-form $\lambda$ ...

**1**

vote

**0**answers

130 views

### Trivialization of holomorphic symplectic 2-form

Let $X$ be a holomorphic symplectic compact manifold with a fixed holomorphic 2-form $\omega$. $\omega$ yields an isomorphism $\phi:T_{X} \rightarrow \Omega_{X}$ via
$$
v \mapsto \phi(v)=\omega(v,-).
...

**33**

votes

**2**answers

2k views

### About a letter by Richard Palais of 1965.

In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read
In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of ...