Hamiltonian systems, symplectic flows, classical integrable systems

**7**

votes

**2**answers

266 views

### When do you go hunting for Lagrangian submanifolds?

Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...

**0**

votes

**0**answers

97 views

### when are the Kahler quotient and the GIT quotient the same? [closed]

Backgroud: Let $(M; J, \omega)$ be a Kahler manifold, $G_\mathbf{C}$ a complex reductive Lie
group acting on $M$ by holomorphic automorphisms and $M\rightarrow \mathfrak{g}^*$ an equivariant
moment ...

**2**

votes

**1**answer

121 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 ...

**0**

votes

**1**answer

83 views

### If a (linear) relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?

Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. Call a relation $U \to V$ a (linear) Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus ...

**1**

vote

**0**answers

59 views

### Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$
acts holomorphically such that the group $K$ preserves the
Kähler ...

**9**

votes

**0**answers

174 views

### Wrapped Fukaya categories of Stein manifolds

By the work of Abouzaid, we know that the wrapped Fukaya category of $T^\ast Q$ with $Q$ a closed smooth manifold is generated by a cotangent fiber. Basically, this is an application of Abouzaid's ...

**6**

votes

**1**answer

162 views

### Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...

**1**

vote

**0**answers

55 views

### Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...

**7**

votes

**1**answer

365 views

### What are the exact holomorphic Lagrangians in complex 2-space?

In an exact symplectic manifold, i.e. where the symplectic form can be written $\omega = d \lambda$, it's natural to look for exact Lagrangians, i.e. $L$ on which $\lambda_L = df$. One reason is ...

**2**

votes

**1**answer

137 views

### Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...

**0**

votes

**0**answers

182 views

### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

**3**

votes

**1**answer

118 views

### Symplectic isotopies between small balls?

Let $(X,\omega)$ be a connected symplectic manifold, possibly with boundary. Let $g_1, g_2: B(1) \to X$ be two balls in $X$. Is it true that if $\delta$ is sufficiently small, then there is an ...

**4**

votes

**1**answer

198 views

### Lagrangian submanifolds in $T^\ast S^n$

Let $L\subset T^\ast S^n$ be a properly embedded Lagrangian submanifold homeomorphic to $\mathbb{R}^n$ with respect to the canonical symplectic structure on $T^\ast S^n$, and suppose $L$ intersects ...

**1**

vote

**1**answer

162 views

### Hamiltonian Isotopy class of Lagrangian Submanifold

Let $(X,\omega)$ be a symplectic manifold, $L\subset X$ be a Lagrangian submanifold, $[L]$ denotes the Hamiltonian isotopy class. How to represent $L'\in[L]$ via $L$ (for example, a graph over $L$)? ...

**2**

votes

**0**answers

39 views

### Casimirs of Poisson brackets obtained via Poisson reduction

Suppose that a Lie group $G$ acts freely and properly on a symplectic manifold $P$ via symplectomorphisms. Suppose further that we have at our disposal an $\text{Ad}^*$-equivariant momentum map ...

**4**

votes

**2**answers

244 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 ...

**13**

votes

**0**answers

278 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**4**

votes

**2**answers

471 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 ...

**2**

votes

**1**answer

344 views

### a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?

**1**

vote

**0**answers

109 views

### A differential equation on toric Kahler manifolds

Let $M$ be a toric Kahler manifold with $\text{dim}_{\mathbb{R}} = 4$. Let $V$ be a Killing vector field associated to the action of the two-torus $\mathbb{T}^2$ on $M$. We also assume the existence ...

**13**

votes

**4**answers

2k views

### Roadmap for Mirror Symmetry

I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side.
I know basic Complex ...

**3**

votes

**2**answers

143 views

### Reference Request: “Neck Stretching Procedure” (In Symplectic Field Theory)

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper ...

**3**

votes

**0**answers

145 views

### Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

I know that many symplectic geometers are interested in quantization as well.
From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in ...

**4**

votes

**0**answers

130 views

### Lagrangian fibration on Schoen's Calabi-Yau 3-fold

Schoen's Calabi-Yau 3-fold is the fiber product $X=Y_1\times_{\mathbb{P}^1}Y_2$ of two rational elliptic surfaces $Y_1\rightarrow\mathbb{P}^1$ and $Y_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and ...

**5**

votes

**1**answer

151 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 ...

**5**

votes

**2**answers

136 views

### Poisson ideals vs. ideals generated by Poisson central elements

Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$:
$I = \langle (c_i) \rangle$ is generated by ...

**1**

vote

**1**answer

96 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 ...

**11**

votes

**0**answers

324 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**1**

vote

**1**answer

196 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 ...

**6**

votes

**5**answers

682 views

### Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...

**4**

votes

**0**answers

94 views

### Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...

**1**

vote

**1**answer

101 views

### Generic deformation of Hilbert scheme of points on K3 surface

i was thinking about deformations of hyperkahler manifolds, in particular hilbert schemes of points on K3 surfaces and I think I realized something. I'm here to ask you if I'm right.
Take $X^{[n]}$ ...

**6**

votes

**1**answer

245 views

### Lifting a Diffeomorphism to the Cotangent Bundle

Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving.
...

**6**

votes

**2**answers

450 views

### Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems).
A ...

**2**

votes

**1**answer

160 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 ...

**0**

votes

**0**answers

205 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

**0**answers

81 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 ...

**11**

votes

**2**answers

1k views

### Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of ...

**6**

votes

**4**answers

575 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} ...

**0**

votes

**0**answers

105 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 ...

**8**

votes

**1**answer

174 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 ...

**2**

votes

**2**answers

278 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 ...

**26**

votes

**1**answer

1k views

### High-Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
Given a ...

**2**

votes

**1**answer

48 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 ...

**3**

votes

**1**answer

172 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 ...

**0**

votes

**1**answer

136 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 ...

**1**

vote

**0**answers

109 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) derivation on $\mathbb{C}[x,y]$:
...

**2**

votes

**3**answers

349 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, ...

**3**

votes

**1**answer

167 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 ...

**7**

votes

**1**answer

440 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 ...