Hamiltonian systems, symplectic flows, classical integrable systems

**8**

votes

**0**answers

98 views

### Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an ...

**0**

votes

**1**answer

121 views

### Volume form on pair (X,D)

Let $X$ be a singular Kahler variety with Kahler current $\omega $ then the volume form is $\omega^n$. Now let $D$ be a divisor then how can we define volume form on pair $(X,D) $?

**1**

vote

**0**answers

111 views

### How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$
with cone singularities along divisor
$D$
with cone angle $2\pi\beta$
is said to be
of
constant scalar curvature Kahler
or
cscK
if its scalar curvature $S(\omega)$, which is ...

**2**

votes

**0**answers

108 views

### Poisson Manifold Structures on Even Dimensional Spheres

The $2n$-sphere, for $n=1,2,3$, possess a (non-trivial) Poisson manifold structure. Is this still true for $n > 3$? Describing the spheres as homogeneous spaces $SO(n)/S(n-1)$, are there Poisson ...

**0**

votes

**0**answers

159 views

### Bigness of a symplectic form on pair $(X,D)$

Let $(M,\omega_M)$ be a compact Kähler manifold. We say that a semi-positive $(1,1)$ form $\omega$ is big iff $$\int_M\omega^n>0$$.
Now let we have the pair $(X,D)$ where $D$ is a divisor on ...

**0**

votes

**1**answer

232 views

### Steps in paper on sympl. geometry unclear

I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.
It deals with the question how the stability properties of a ...

**1**

vote

**0**answers

69 views

### Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...

**3**

votes

**1**answer

127 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**3**

votes

**1**answer

97 views

### Action generated by geodesic flow is hamiltonian

I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle ...

**4**

votes

**1**answer

172 views

### Check symplectomorphism property on infinitesimal generators

I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G ...

**0**

votes

**0**answers

129 views

### Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be ...

**2**

votes

**2**answers

170 views

### Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again ...

**2**

votes

**1**answer

149 views

### Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
The background
So he wants to show that any symplectic form is ...

**1**

vote

**0**answers

129 views

### Symplectic spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry.
The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le ...

**1**

vote

**0**answers

71 views

### Limiting behaviour of the symplectic form

It is well known that the coadjoint orbits of the Heisenberg group (with a suitable choice of coordinate system) are the planes $z=c\ne 0$ parallel to the $xy$-plane, and the points in the $xy$-plane. ...

**3**

votes

**0**answers

42 views

### multiplicity free chain of Lie subgroups gives a completely integrable system on symplectic manifold

Assume we have a symplectic manifold $N$ and a Hamiltonian action of a Lie group $G$, such that the algebra of all $G$-invariant functions on $N$ is commutative under Poisson bracket.
Furthermore, ...

**1**

vote

**0**answers

81 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**1**

vote

**0**answers

146 views

### Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...

**0**

votes

**1**answer

155 views

### Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...

**0**

votes

**0**answers

55 views

### Characterisation of vector fields solution to a simple equation

This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ...

**0**

votes

**1**answer

149 views

### When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?

Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $
a fixed homology class that is $\textit{decomposable}$. Let
$$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$
denote the stable ...

**0**

votes

**0**answers

75 views

### spectral sequence in equivalent Floer cohomology

Let $M$ be a symplectic aspherical manifold with Hamiltionian group action by $G$.
Under suitable assumptions, one can define the equivalent floer cohomology $FH^*_G(M)$ by
using symplectic vortex ...

**0**

votes

**0**answers

141 views

### How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?

This is a very basic question about the definition of Moduli space of maps.
My reason for asking this question is because I haven't actually seen this
definition explicitly given anywhere, and hence ...

**1**

vote

**0**answers

348 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics ...

**0**

votes

**0**answers

111 views

### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

**6**

votes

**1**answer

462 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

**2**

votes

**1**answer

169 views

### Boundary conditions and the relationship between Hamiltonian and Lagrangian Floer theories

I am looking for a full account of the relationship between the various versions of Floer theory on a symplectic manifold $M$. If we take the usual Floer equation (Hamiltonian version)
...

**0**

votes

**1**answer

156 views

### Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...

**20**

votes

**2**answers

682 views

### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...

**5**

votes

**1**answer

214 views

### Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**1**answer

205 views

### What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**4**

votes

**2**answers

186 views

### actual dimension of concrete moduli space of holomorphic curves vs its virtual dimension

I am looking at exercise 6.3.3 in Mcduff's and Salamon's book J-holomorphic curves and Symplectic topology, which basically gives an example of a moduli space whose actually dimension is greater than ...

**3**

votes

**1**answer

116 views

### neighborhood of symplectic surfaces

I want to know if there is a uniqueness (in any sense) theorem for the symplectic structure in a neighborhood of a symplectic surface in a four dimensional symplectic manifold. Or more generally for ...

**5**

votes

**1**answer

202 views

### Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing ...

**4**

votes

**1**answer

233 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...

**2**

votes

**2**answers

144 views

### Hamiltonian group actions in the context of holomorphic line bundles

When studying Hamiltonian group actions, a very nice set up might be to take the following:
Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian ...

**9**

votes

**0**answers

105 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...

**0**

votes

**1**answer

99 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**2**

votes

**1**answer

98 views

### AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?

**1**

vote

**1**answer

237 views

### symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?

**0**

votes

**0**answers

66 views

### Convenient Basis Presentation of Lefschetz Decomposition

Let $V$ be an almost-complex vector space, equipped with a symplectic element $\omega \in V^{(1,1)}$. In terms of a basis $b^+_i \in V^{(1,0}$, $b^-_i \in V^{(0,1}$, does there exist a "simple" ...

**12**

votes

**2**answers

314 views

### Is there a relationship between Fourier transformations and cotangent spaces?

Maybe a trivial question but I can't seem to find it treated anywhere.
Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function ...

**0**

votes

**0**answers

73 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

**4**

votes

**1**answer

113 views

### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...

**3**

votes

**0**answers

181 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, ...

**2**

votes

**0**answers

95 views

### Generation of compact Lagrangians over fields with characteristic 2

Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...

**2**

votes

**0**answers

81 views

### Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional ...

**3**

votes

**2**answers

181 views

### Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...

**3**

votes

**1**answer

114 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

**1**

vote

**1**answer

190 views

### Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 ...