Hamiltonian systems, symplectic flows, classical integrable systems

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63 views

### What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...

**0**

votes

**0**answers

53 views

### About some 'rigidity theorem' for the Kahler forms on projective bundles

Let $E\to X$ be a holomorphic vector bundle over a compact Kahler manifold $X$ with Kahler form $\omega_{X}$. For a given hermitian metric on $E$, let $\omega_{E}$ be the Chern form of the line bundle ...

**2**

votes

**1**answer

55 views

### Embedded Contact Homology and Manifold Decompositions

Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.
In the symplectic ...

**2**

votes

**0**answers

67 views

### Gromov width of cotangent disk bundle

Given a symplectic manifold $(M^{2n},\omega)$, the Gromov width of $M$ is defined to be
$w(M)=sup\{{\pi r^2| B^{2n}(r) \rightarrow M}\}$
My question is: what is the explicit value of $w(D^*S^n)$, ...

**1**

vote

**0**answers

64 views

### Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$

I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...

**1**

vote

**0**answers

69 views

### Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...

**0**

votes

**0**answers

49 views

### computation of floer homology of cotangent bundle of spheres

I am wondering the computation of the floer homology of cotangent bundle of spheres.
By a theorem of Viterbo, it is isomorphic to the homology of free loop space of sphere.
However, I am wondering ...

**4**

votes

**0**answers

84 views

### Existence of a torus fibration with given vanishing cycles

Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be ...

**0**

votes

**1**answer

87 views

### symplectic homotopy equivalence

Let $(M,\omega)$ and $(M^{\prime},\omega^{\prime})$ are two symplectic manifolds. Then we may define a natural homotopy equivalence as follows.
We say that the smooth map $f:M\longrightarrow ...

**0**

votes

**0**answers

14 views

### Condition for local Lipschitzness of pullback map for exterior forms

Given $w\in\Lambda^k(\mathbb{R}^n)$, determine the condition under which the map $T\rightarrow T^*(w)$ is locally Lipschitz, where $T\in GL_n(\mathbb{R})$ and $T^*(w)$ denotes the Pullback of $w$ by ...

**1**

vote

**1**answer

166 views

### Symplectic and Holomorphic Vector Bundles

As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the ...

**2**

votes

**0**answers

61 views

### Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation
$$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$
can ...

**2**

votes

**1**answer

297 views

### Kodaira dimension of co-adjoint orbit

Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...

**1**

vote

**0**answers

147 views

### Dropping the closed requirement from the symplectic manifold definition?

A symplectic manifold is a pair $(M,\omega)$, where $\omega$ is a non-degenerate closed two-form. When $M$ is compact, Hodge decomposition implies that such manifolds have non-zero second ...

**0**

votes

**0**answers

75 views

### Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is:
Is the group of symplectic automorphisms
of $S^2$ with respect to this ...

**3**

votes

**1**answer

145 views

### Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...

**2**

votes

**0**answers

182 views

### Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
...

**2**

votes

**0**answers

116 views

### Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it.
Is there any known construction of a bundle with a ...

**10**

votes

**6**answers

542 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

**1**answer

88 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

65 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

319 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 ...

**1**

vote

**0**answers

67 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 ...

**6**

votes

**1**answer

179 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 ...

**7**

votes

**1**answer

434 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 ...

**0**

votes

**0**answers

193 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 ...

**2**

votes

**0**answers

45 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 ...

**1**

vote

**1**answer

182 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 $L$)? ...

**13**

votes

**0**answers

300 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 ...

**1**

vote

**0**answers

114 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 ...

**3**

votes

**2**answers

169 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

172 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

147 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

**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 ...

**11**

votes

**0**answers

348 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 ...

**4**

votes

**0**answers

102 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 ...

**4**

votes

**1**answer

231 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 ...

**4**

votes

**1**answer

209 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 ...

**3**

votes

**1**answer

125 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 ...

**1**

vote

**1**answer

104 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

261 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.
...

**2**

votes

**1**answer

164 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

203 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

156 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

114 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

297 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

49 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

130 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 $\mathbb{C}[x,y]$:
...