Hamiltonian systems, symplectic flows, classical integrable systems

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298 views

### Does symplectic homology admit a natural Batalin-Vilkovisky algebra structure?

[Could someone please change this to a motivating paragraph instead of just repeating the title? - S.C.]

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35 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 ...

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256 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 ...

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37 views

### Canonical relations and phase functions of a Fourier Integral Operator

I'm thinking about the (semiclassical) Fourier Integral Operator $T$ given by
$T=h^{-n}\int{e^{i\phi(x,y,\theta)/h}a(x,y,\theta,h)d\theta}$
(that is, $T$ has phase $\phi$ and amplitude $a$). ...

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**1**answer

375 views

### Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.
My question is as ...

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79 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 ...

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**1**answer

105 views

### model compact coisotropic submanifold

$\{x_1,\ldots,x_n,\xi_{m+1},\ldots,\xi_n\in\mathbb{R},\xi_1=\ldots=\xi_m=0\}$ is a "model" codimension $m$ coisotropic submanifold of $T^*\mathbb{R}^n$, and is of course noncompact. Is there such a ...

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259 views

### Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...

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760 views

### Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models:
It is an important player in the story of mirror symmetry.
It involves "potentials" which are functions of complex varibles, which have isolated ...

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**1**answer

28 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 ...

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139 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 ...

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189 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, ...

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135 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 ...

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129 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 $n-2$ ...

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186 views

### Why non-compact Calabi-Yau surfaces are not self-mirror?

By the work of Gross and Bernard-Matessi, in dimension 3 $T$-duality should be understood as an exchange of positive and negative local model of Lagrangian torus fibrations, at least in its ...

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196 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

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143 views

### Local holomorphic equations for symplectic divisors

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $X$ admits a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of ...

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678 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 $ ...

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150 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, ...

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53 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 ...

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268 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits

Edit: There is an interesting complete answer for the second part(see the answer by Thomas Kragh). I search for an answer for the first part.
1.Is there a polynomial Hamiltonian ...

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63 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?

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111 views

### How to compute Conley-Zehnder indices on prequantization spaces?

My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits ...

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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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81 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 ...

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515 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} ...

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231 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 ...

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**1**answer

160 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 ...

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145 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 ...

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250 views

### Is there a Legendrian Neighbourhood Theorem also for non-cooriented contact manifolds?

Context
According to Arnol'd, a contact structure on a smooth manifold $M$ is given by a corank 1 tangent distribution $C$ which is maximally non-integrable; this means that, for any local $1$-form ...

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145 views

### Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration?

Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold ...

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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 $ \# ...

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161 views

### Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...

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101 views

### A question about solutions to Floer's equation which are asymptotic to a stationary point

Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X_H\otimes ...

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202 views

### computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...

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80 views

### Detecting Monodromy in Integrable Systems

Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know ...

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444 views

### Maslov index and heegard floer homology

I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...

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124 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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891 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). ...

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109 views

### Unobstructed Lagrangian tori

Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in ...

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78 views

### wavefront is a coisotropic

I'm thinking about the following example: Let $u_k:=e^{i(kx_1+k^2x_2)}$, where $k\in\mathbb{N}$ and $x_1,x_2\in\mathbb{R}$. Then for the sequence $h_k:=1/(k\sqrt{1+k^2})$ ($h_k\rightarrow0$ as ...

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108 views

### Question on Ionel and Parker's paper: Relative Gromov Witten Invariants

In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ...

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405 views

### Spaces of symplectic embeddings: Bundle? Smoothness?

Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth ...

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175 views

### coisotropic submanifolds

I'm thinking about coisotropic/involutive submanifolds of the symplectic phase space $T^*\mathbb{R}^n$ (with coordinates $x_1,\ldots,x_n,\xi_1,\ldots,\xi_n$). As I understand, the smallest ...

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534 views

### Kenji Fukaya's Lecture series at Simons center

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry.
...

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66 views

### about energy bound in Fukaya category

In Fukaya category, moduli spaces is defined, which are solutions of certain $C$-$R$ equations, which involve strip ends in boundary condition. When the number of strip ends $>2$, a curvature term ...

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169 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

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120 views

### Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...

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301 views

### contactomorphism of $S^{2n+1}$ for n>1

Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1?
For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the ...

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94 views

### Classification of Compact Symplectic Homogeneous Spaces

Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...