Hamiltonian systems, symplectic flows, classical integrable systems

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5
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1answer
270 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
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 ...
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 ...
0
votes
0answers
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
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
votes
1answer
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
2answers
507 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
3answers
781 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
1answer
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
1answer
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
1answer
603 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
0answers
196 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
1answer
189 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
1answer
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
0answers
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
1answer
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
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2answers
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
2answers
273 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
2answers
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
1answer
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
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
vote
1answer
252 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
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
votes
2answers
888 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
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0answers
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
1answer
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
2answers
262 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
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 ...
4
votes
0answers
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
1answer
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
1answer
219 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
4answers
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
1answer
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
5answers
671 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
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0answers
140 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
2answers
413 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
1answer
784 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
1answer
290 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
2answers
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
3answers
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
1answer
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
2answers
711 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
1answer
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
2answers
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
1answer
416 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 ...
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0answers
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
2answers
426 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
1answer
376 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
0answers
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,-). ...
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votes
2answers
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 ...