# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,258
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### How much further could the PSS morphism be pushed?

Given a closed symplectic manifold $(X, \omega)$, the well known Piunikhin-Salamon-Schwarz morphism identifies the quantum cohomology $QH(X, \omega)$ with the Hamiltonian Floer cohomology $HF(X,H)$ ...

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### $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology

Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$...

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51 views

### Hyperkahler and symplectic complex geometry: reference?

I would need some references regarding symplectic and hyperkahler (complex) geometry. My background is mostly from algebraic geometry and I know a little bit the basics on Kahler manifolds.
I would be ...

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114 views

### Coadjoint orbits

I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer!
While I was trying to teach my ...

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35 views

### Classification of genus 1 Lefschetz fibration over a disk with 6 singular fibers

For a genus 1 Lefschetz fibration over a sphere, there is a good classification in B. Moishezon, Complex surfaces and connected sums of complex projective planes. In this case, the number $K$ of ...

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238 views

### algebraic momentum map

Let $T$ be a linear algebraic torus over $\mathbb C$ and $X$ be a smooth quasi-projective symplectic $T$-variety. Also, assume that the action of $T $ is free and $X/T$ exists as a smooth variety. Is ...

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### What is so geometric about symplectic geometry?

Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't ...

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### Relationship between exact symplectomorphisms and Hamiltonian diffeomorphisms (in the context of Milnor fibers)

Here is a preamble/setup. Suppose we have a symplectic manifold $(M,d\lambda)$ with an exact symplectic form. By Stokes theorem, $M$ must have nonempty boundary. An exact symplectomorphism $\phi:M \to ...

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201 views

### Cotangent bundles of surfaces as varieties

As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z_1^2+z_2^2+z_3^2=1$$
with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...

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120 views

### Composition of coproduct and product in Lagrangian Floer (co)homology

Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...

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### An analogue of the Poisson bracket in contact geometry?

I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...

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### Are the odd dimensional spheres Poisson homogeneous spaces?

Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?

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162 views

### Are there known examples of almost complex manifolds admitting neither a symplectic nor a complex structure? [closed]

I have seen the the example of $S^6$ being touted around here and there but it does not seem to be generally confirmed that there is no complex structure on it.

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### Choice of almost complex structure in Seidel's Symplectic Floer Homology of a Dehn twist

I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for ...

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445 views

### Every half-dimensional subspace of a symplectic vector space has a Lagrangian complement

Let $(V, \omega)$ be a finite-dimensional real symplectic vector space, i.e. $\omega : V \times V \to \mathbb{R}$ is a non-degenerate skew-symmetric bilinear map.
A linear subspace $L \subset V$ is ...

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### Integrable systems and Lagrangian fibrations

It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...

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### Relating the Morse index with the Maslov index

In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...

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### Upper triangular similitude for symplectic matrices

It is known that given any matrix $M$ in $Sp(2,\mathbb{Z})$ with eigenvalue $+1$, we can find a real symplectic matrix $S$ such that $S^{-1}MS$ is upper triangular with diagonal entries equal to $+1$.
...

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### Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds

I started reading the book "Lagrangian intersection Floer theory anomaly and obstruction", and there are a couple of details and assumptions in the definition of the Novikov covering that I ...

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### Existence of uniformly bounded Darboux chart

In Donaldson's paper Symplectic submanifolds and almost-complex geometry, he mentioned that for each point $p$ in a compact almost-Kähler manifold $(V,\omega ,J)$, there exists a Darboux chart $\...

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### Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds

Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a ...

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277 views

### Symplectic resolutions amongst cotangent bundles

It is known that a generalized flag variety $X=T^*(G/P)$ is a (symplectic) resolution of singularities of its affinization $X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$. In type ...

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### Compactness as a consequence of the adjunction formula for genus second homology class

Recall the adjunction formula $$ g(\alpha) = 1 + \frac{1}{2}\left( \alpha^2 -c_1(X)\cdot \alpha \right)$$ where $g(\alpha)$ is the genus of a pseudoholomorphic representative of the Poincaré dual of $\...

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### Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !

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### A theorem about the symplectic geometry of projective bundles

I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta ...

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158 views

### Hamilton equations-Symplectic scheme [closed]

We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...

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### Maslov cycle for the Conley-Zehnder index - what are its regular points?

I'm looking at the definition of the Conley-Zehnder index, where it is important to look at the group $$\text{Sp}(2n)^* := \{ A \in \text{Sp}(2n) | \det (A-\text{Id}) \neq 0 \}$$and its complement $$\...

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### Dynamics of fiberwise starshaped hypersurface of Hamiltonian flows on $T^*M$

I have started reading the following paper arXiv link on Dynamical Systems and Symplectic Geometry and in page $3$ we have the following statement :
Let $\Sigma$ be a fiberwise starshaped ...

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237 views

### How should we think about the algebraic moment map?

My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...

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### Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?

Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space.
According to Gromov's h-principle for contact embeddings, any contact manifold ...

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388 views

### Arnold's book on classical mechanics [duplicate]

Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...

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### Action functional for the definition of Lagrangian Floer homology

I have been starting to learn about Lagrangian Floer homology using notes by A. Pedroza (arXiv link).
Consider $(M,\omega)$ a symplectic manifold that is symplectic aspherical and $L_0,L_1$ two ...

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### Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold

Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...

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### Does quantum cohomology have an $E_\infty$-ring structure?

Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication ...

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### Fixed point set of diffeomorphism is a submanifold

I am in the following setting: Let $(\Sigma,\alpha\vert_\Sigma)$ be a compact regular energy surface of restricted contact type in an exact Hamiltonian manifold $(M,d\alpha,H)$. Given $\varphi \in \...

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### Why is embedded contact homology so powerful?

The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in ...

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### symplectic Runge-Kutta for matrix differential equation

I would like to solve, for $t>0$ the following matrix differential equation:
$$U'(t)=H(t)U(t)$$
with initial condition $U(t=0)=U_0$ ($2N\times2N$, symplectic and unitary matrix) and $H(t=0)=H_0$ ($...

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### $2$-Form inducing a non-degenerate form on $\Gamma(T\mathbb{R}^{2n+1})$

Every $2$-form $\omega\in \Omega^2(\mathbb{R}^{2n+1})$ induces a skew-symmetric map
$$
\omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1})
$$
...

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### Stabilizer groups of Yang-Mills connections

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.
...

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### How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...

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### Deform a complex structure fixing marked points

Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism ...

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### Lagragian floer homology vs homology of $\Omega(L_0,L_1)$

I'm very new to this subject, so apologies for a very naive question and probably many mistakes. Let $M$ be some compact sympletic manifold with $L_0,L_1$ Lagrangian submanifolds which intersects ...

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### Regularity of the dependence of the flow on the vector field definining it

Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...

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### Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$

I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...

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### Is every $M\in \mathfrak{s}\mathfrak{p}_4(F)$ conjugate to an “upper triangular” matrix?

Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the ...

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### Symplectic form on $\Omega^0(X,End(E))$

Let $E\rightarrow X$ be a holomorphic vector bundle over a Kahler manifold. Is there a natural symplectic form on the space $\Omega^0(X,End(E))$ ?
For example on $\Omega^1(X,End(E))$ we have the ...

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### Is composition of discrete Hamiltonian flows integrable?

Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$
For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...

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### What is the significance of a Lagrangian Submanifold and what are the implications of the symplectic form being zero?

I'd like to understand better the relevance of Lagrangian submanifolds in Hamiltonian Mechanics. A Lagrangian Manifold is defined as a submanifold of a symplectic manifold upon which the restriction ...

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### Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...

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181 views

### Moduli space of annuli with marked points satisfying some additional symmetries

Let us consider the space of configurations $\overline{\mathcal{M}}_{0,2,1,(1,1)}$ of an annulus with a marked point on the interior boundary component (let's call it "out") a marked point ...