Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

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Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
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Darboux's Theorem Alternate Proof

I've been given the task of proving Darboux's theorem through non-standard means. Definitions Let $(M,\phi)$ be a symplectic manifold. $\mathcal{F}_{\text{SP}(V)}(M)$ is the bundle of frames $(\eta^...
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When is a vector field hamiltonian with respect to some symplectic form?

Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...
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59 views

Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
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rigid body poisson bracket

i have trouble to understand the definition of the rigid body poisson brackets. In the book of Marsden and Ratiu "Introduction to Mechanics and Symmetry", in chapter 10.1 they introduce the poisson ...
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Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, $\...
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118 views

Weyl orbits of integral dominant weights and convex polytopes

Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an $n$-...
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107 views

Scattering of waves by an obstacle

I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. http://www.sciencedirect....
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349 views

basis free volume form for a symplectic vector space

It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$ (see for example Jason DeVito's excellent answer to ...
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65 views

Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a complex vector space together with fixed symplectic ...
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30 views

How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter $\...
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59 views

A coisotropic submanifold is locally given by the fibers of a submersion with coordinates in involution

Let $(M, \omega)$ be a symplectic manifold and $Q \subset M$ a coisotropic submanifold of codimension $k$. I'm trying to prove that for every $x \in Q$, there exists an open subset $U \subset M$ ...
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42 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
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43 views

Pulling back a Kähler structure on a symplectic submanifold

Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
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Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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What is the actual definition of “polarizing form”, in the context of cohomology algebras?

I am studying Voisin's Hodge Theory and Complex Algebraic Geometry, in order to better understand the underpinnings of her 2008 paper on Hodge structures. She discusses elements $\omega$ in $H^{2}(...
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92 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
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Help with constructing a symplectic form on a $2-$dimensional torus

Consider the standard two dimensional torus $M=\mathbb{R}^2 / \Gamma$ where $$ \Gamma:=\{(n,m)\in \mathbb{R}^2:n,m\in \mathbb{Z}\}. $$ I want to $1)$ construct a symplectic form $\omega$ on $M$, ...
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108 views

About $\operatorname{Spin}^\mathbb{C}$ structures

Let $X$ be a orientable 4-manifolds. I know that $X$ can be endowed with $\operatorname{Spin}^\mathbb{C}$ structures by the choice of integral lift of $w_2(X)$ (second Stiefel-Whitney class of ...
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Question about definition of non-compact Calabi-Yau manifolds

Using the following definition: Definition $(X, J, \omega, \Omega)$ is a Calabi-Yau manifold if $g(\cdot, \cdot)= \omega(\cdot, J \cdot)$ and $\Omega$ is a nowhere vanishing holomorphic $(n,0)$-form ...
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Coordinates for a neighborhood of a totally real submanifold

Motivation: I'm interested in which submanifolds of (half dimension) a complex manifold can be expressed locally in coordinates as the real part of the complex coordinates, and was wondering if the ...
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Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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Level sets and lagrangian submanifolds

I would like to find all regular values for a map on a symplectic manifold such that the level sets are lagarangians. Precisely, the following example : The $4$-dimensional symplectic manifold $(\...
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A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
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33 views

What's the best place to learn Quantum Homology

I am a PhD student and I am trying to learn quantum homology. I already know some of the analysis, but I am struggling to really find a good readable reference which covers enough of the analysis to ...
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39 views

Computing real de Rham cohomology of Hironaka's 3-manifold example

I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two ...
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49 views

Showing Hofer's metric is bi-invariant

Im trying to show that the Hofer metric on a symplectic manifold is bi-invariant but im struggling. Firstly, given the flow $\rho_t$ of a hamiltonian $H_t$, the Hofer metric is $$ d(\rho_0,\rho_1)=\...
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Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
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Question on symplectic geometry

I am currently reading this paper on symplectic geometry. It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second ...
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38 views

Any good resources for learning about the moduli space of symplectic structures on a given manifold?

What I can find on the subject are the papers by Fricke and the Habermanns: http://www.researchgate.net/publication/227336993_On_the_geometry_of_moduli_spaces_of_symplectic_structures and http://www....
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The meaning of variables and derivations in Souriau's book

As far as I see, Souriau is using unconventional notions in his book "Structure of Dynamical Systems". He explains these notions in §2. of Chapter I, but it is a puzzle for me. Mainly because he ...
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Non-commutative symplectic geometry

How is non-commutative symplectic geometry defined? How does it differ from symplectic geometry? Does Darboux's theorem apply also there?
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Finding a path of symplectic forms

Suppose $M$ is a symplectic manifold and $\omega_0$ and $\omega_1$ are two symplectic forms belonging to the same de Rham cohomology class. Assume in addition they can be connected via a path of ...
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Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} &&dd^*(\frac{r}{8}(1-|\alpha|^2+|\beta|^2)...
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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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Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension $n$...
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Trivialization of a path of tamed almost complex structures

I am wondering if the following result is true: Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by $\...
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Area of flux homomorphism in symplectic topology

Let $(M,\omega)$ be a symplectic manifold. Let $f:M \to \mathbb R$ be a smooth function. We have vector fields $X_f$ defined by $\omega(X_f,)=df$. Let $\phi_t$ be the flow of $X_f$ and let $\gamma: \...
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Index of a Symmetric Matrix

In Hansjorg Geiges' introductory textbook on Symplectic Geometry, is defined a projective conic given by $q^tAq=0$ where $A$ is a symmetric matrix of rank 3 and index 2. What does "index 2" mean? I ...
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Geodesic Flow is the flow of the Hamiltonian Vector field of $|\xi|$

Let $M$ be a complete Riemannian manifold with metric $g$. The geodesic flow is the one-parameter family of diffeomorphisms $\phi_t$ on $T^*M$ defined as follows. If $\xi \in T^* M$ is a vector based ...
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How to prove that a certain action is hamiltonian?

Reading a paper I had the need to complete a proof, and come up with a certain argument(see below). My question is: could I reduce it to a special case of some theorem? I ask this question in order to ...
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Geodesic flow generated by Riemannian distance function

This is an exercise in AC da Silva's Lectures onn Symplectic Geometry; I am having trouble showing the following. $(X,g)$ is a geodesically complete manifold, and $f: X \times X \to \mathbb{R}$ is ...
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decomposition of general symplectic matrix

Suppse $S$ be a $2n \times 2n $ symplectic matrix $$ S = \left( \begin{array}{cc} A & B\\ C & D \end{array} \right) . $$ By definition, $S^tJS$=J, where $$J = \left( \begin{array}{cc} 0 &...
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Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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Pushforward of Liouville measure on configuration space

Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the ...
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Additional References — Symplectic Geometry

Please forgive me if I should ask this question somewhere else. If so, let me know and I will do so. I have never asked a question of this nature before, and so am unsure where an appropriate place ...
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Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
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Infinite dimensional Hamiltonian systems: looking for textbook/general results

Consider an infinite-dimensional phase space $(X,\omega)$, where $X = V \times V'$ with $V$ being a Banach space and $\omega$ a (weak) symplectic form. Let $E : X \to \mathbb{R}$ be a smooth function, ...
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How does Melnikov function for a Hamiltonian change if one considers an augmented symplectic manifold?

Suppose we have a nonautonomous nearly integrable Hamiltonian system, periodic in $t$ with period $2\pi / \omega$ $$H_{\epsilon}(x,y,t)=H_{0}(x,y) + \epsilon H_{1}(x,y,t)$$ with $(x,y,t) \in \mathbb{...
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When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...