Hamiltonian systems, symplectic flows, classical integrable systems

**2**

votes

**1**answer

167 views

### How to find Darboux coordinates?

I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be ...

**1**

vote

**1**answer

206 views

### Computation with the Legendre Transform

Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x_1,\dots x_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates ...

**2**

votes

**0**answers

82 views

### When is a symplectomorphism on a cotangent bundle the lift of a diffeomorphism on the base manifold [duplicate]

Let $X$ be a manifold and $T^\ast X$ the cotangent bundle. Let $\alpha$ denote the tautological $1$-form on $T^\ast X$ so that $(T^\ast X, \omega=-d\alpha)$ is a symplectic manifold. I want to know ...

**5**

votes

**1**answer

157 views

### Symplectic Koopmanism

Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville ...

**0**

votes

**0**answers

118 views

### Gromov-Floer compactness for C^0 convergence of complex structure/ C^1 convergence of Hamiltonian

Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of ...

**2**

votes

**1**answer

131 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 ...

**2**

votes

**2**answers

307 views

### Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps?

Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J_N$ and $J_M$).
Does it imply ...

**2**

votes

**1**answer

52 views

### Kostant-Kirillov form versus Fubini-Study form on Plucker embedding

Let $G$ be a connected complex reductive group with a maximal compact subgroup $K$.
Let $\lambda$ be a dominant weight in the interior of the positive Weyl chamber. Let $V_\lambda$ denote the ...

**1**

vote

**0**answers

132 views

### The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$:
...

**2**

votes

**3**answers

354 views

### If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which
acts on $M$ and preserves complex structure then If $M$ has Kaehler
manifold, then the symplectic quotient of $M$, i.e, ...

**7**

votes

**1**answer

459 views

### Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know
Let $(M,\omega)$ be a compact ...

**8**

votes

**2**answers

248 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 ...

**3**

votes

**1**answer

174 views

### quantum states and observables for the non-commutative torus

The noncommutative torus $A_\theta$ is a $C^*$-algebra corresponding to an irrational foliation of the torus $\mathbb{T}^2$ by lines of slope $\theta \notin \mathbb{Q}$.
As far as I am reading it is ...

**1**

vote

**1**answer

164 views

### When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and
$\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...

**1**

vote

**0**answers

176 views

### Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$
I am looking for a proof for ...

**4**

votes

**2**answers

255 views

### Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...

**1**

vote

**1**answer

310 views

### A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, ...

**4**

votes

**0**answers

124 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 = ...

**11**

votes

**1**answer

347 views

### What is Known about the $K$-Theory of Fukaya Categories?

Some Background: In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed ...

**1**

vote

**1**answer

97 views

### Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities".
Let $f\colon \mathbb{C}^n\to ...

**3**

votes

**1**answer

179 views

### Displaceability of submanifolds

My question is motivated by the following question.
How transitive are the actions of symplectomorphism groups ?
A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there ...

**7**

votes

**1**answer

218 views

### Formula for the Haar measure in the linear symplectic group

What is (or where can I find) an explicit formula for the Haar measure of the group of linear symplectic transformations of $\mathbb{R}^{2n}$?
Added 13/05/2014.
Some clarifying remarks:
(1) by ...

**2**

votes

**0**answers

173 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

**1**answer

197 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 ...

**3**

votes

**2**answers

227 views

### Almost Toric Symplectic Four-Manifolds

Let $(M,\omega)$ be a closed, symplectic four-manifold admitting an almost toric fibration, in the sense of Symington and Leung (e.g. http://arxiv.org/pdf/math/0210033.pdf). That is, there is a ...

**3**

votes

**1**answer

123 views

### use Floer homology to prove the fixed points

I read paper, in page 21, there is a proposition:
Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0$ = id, $f_1= f$ be a Hamiltonian path on M generated by a ...

**1**

vote

**0**answers

49 views

### About interpolability of Stein structures

Imagine $V$ is a complex vector space and $U_1\subset U_2\subset V$ two Euclidean balls.
Let $\psi,\psi_1$ be strictly plurisubharmonic functions on $V$ and $U_1$ respectively.
Question: What are ...

**3**

votes

**1**answer

138 views

### All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...

**2**

votes

**0**answers

187 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

**1**

vote

**2**answers

117 views

### Chern classes of reduced space for Hamiltonian circle action

I have a question about Chern class of symplectic reduction.
Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action.
Let $H : M \rightarrow \mathbb{R}$ be the ...

**1**

vote

**1**answer

149 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 ...

**4**

votes

**2**answers

589 views

### A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...

**0**

votes

**1**answer

198 views

### Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...

**4**

votes

**0**answers

100 views

### Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...

**3**

votes

**0**answers

83 views

### DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...

**3**

votes

**0**answers

115 views

### Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...

**1**

vote

**1**answer

45 views

### Reduction along an Orbit for C.-M. systems

I am having trouble in understanding the section of this paper http://www-math.mit.edu/~etingof/zlecnew.pdf
where the author introduces the Calogero-Moser system as the reduction of a manifold $M$ on ...

**2**

votes

**1**answer

270 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

**7**

votes

**1**answer

311 views

### When a symplectic manifold is formal?

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, then we have the symplectic Hodge operator
$$*:\Omega^{k}(M)\rightarrow\Omega^{2n-k}(M)$$
Furthermore, we can define a differential ...

**0**

votes

**0**answers

160 views

### Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...

**2**

votes

**1**answer

183 views

### Extending Reeb field from contact submanifold to ambient contact manifold

Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...

**2**

votes

**1**answer

143 views

### Anti_symplectic 2-forms

A two form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T_{x} M \mid i_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T_{x}M$. So we obtain a $n-2$ ...

**2**

votes

**0**answers

80 views

### Is it true that a nondegenerate minimizing periodic orbit of mechanical Hamiltonian system is hyperbolic

Consider mechanical Hamiltonian system of the form
$$H(p,q)=\dfrac{\Vert p\Vert^2}{2}+V(q),\quad (q,p)\in T^*\mathbb T^n.$$
Here we suppose the periodic orbit $\gamma$ minimizes the Lagrangian ...

**3**

votes

**0**answers

212 views

### Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...

**0**

votes

**0**answers

103 views

### filling by holomorphic disks method

Can you give me a reference for the proof of the filling by holomorphic disks method, besides Bishop's original paper?

**29**

votes

**1**answer

790 views

### What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...

**2**

votes

**1**answer

113 views

### Computation of symplectic quasi-state

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a ...

**1**

vote

**1**answer

272 views

### A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...

**2**

votes

**1**answer

184 views

### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...

**1**

vote

**1**answer

156 views

### The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...