# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,258
questions

**11**

votes

**2**answers

390 views

### “Sameness” of dg and A-infinity categories

Let $k$ be a field.
A folklore theorem states that dg-categories (over $k$), $A_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example
...

**2**

votes

**1**answer

104 views

### $\mathbb{R}\mathbb{P}^n$ 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 $\mathbb{C}\mathbb{P}^n$...

**1**

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**0**answers

41 views

### 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 $HF(X,H)$ ...

**12**

votes

**3**answers

3k views

### Question about Hodge number

Hi. I am studying Hodge theory on Kahler manifolds.
I have several questions.
Is Hodge number a topological invariant? (I mean, is it independent of the choice of
Kahler structure?)
If the question ...

**2**

votes

**1**answer

239 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 ...

**2**

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**0**answers

56 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 ...

**16**

votes

**2**answers

873 views

### Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?

This question is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth area-preserving diffeomorphism $f:D \to D$ that does not have conformal points?
I ...

**1**

vote

**0**answers

115 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 ...

**1**

vote

**0**answers

36 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 ...

**37**

votes

**3**answers

2k views

### 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 ...

**2**

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64 views

### 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 ...

**8**

votes

**1**answer

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 ...

**6**

votes

**0**answers

121 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 ...

**1**

vote

**1**answer

134 views

### 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 ...

**0**

votes

**1**answer

145 views

### Diameter of pseudoholomorphic curves

Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk.
My question: can one prove an a-priori bound on the diameter of $u$ (...

**2**

votes

**0**answers

48 views

### 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?

**3**

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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.

**4**

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105 views

### 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 ...

**6**

votes

**2**answers

447 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 ...

**11**

votes

**3**answers

2k views

### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

**4**

votes

**2**answers

440 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 $\...

**1**

vote

**0**answers

124 views

### Confusion about the definition of a formal Legendrian isotopy

We say two Legendrian embeddings $f_0,f_1:L^n\rightarrow (Y^{2n+1},\xi)$ are formally isotopic if there is a smooth isotopy $f_t$ connecting $f_0$ and $f_1$ and a bundle monomorphism $F_t^s:TL\...

**3**

votes

**2**answers

1k views

### Legendrian Tubular Neighborhood Theorem

Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian ...

**3**

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**0**answers

167 views

### Coulomb branch varieties and symplectic singularities

I was recently looking at the survey article of Fu on symplectic resolutions which has a number of open questions and conjectures at the end. (I think one of these was existence of a classification ...

**2**

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**0**answers

67 views

### 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 ...

**1**

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**0**answers

55 views

### 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 ...

**7**

votes

**1**answer

319 views

### Large isometry groups of Kaehler manifolds

Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...

**2**

votes

**1**answer

296 views

### A $C^2$ small autonomous Hamiltonian has only constant 1-periodic orbits

Consider a autonomous Hamiltonian $h:W\rightarrow \mathbb{R}$, where $W$ is a symplectic manifold. Let $\mathrm{sgrad}(h)$ denote the vector field on $W$ that is dual to the differential $Dh$ using ...

**1**

vote

**0**answers

43 views

### 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 $+1$.
...

**3**

votes

**1**answer

362 views

### Geometric invariants of a Riemannian manifold encoded in certain moment map

Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...

**3**

votes

**1**answer

146 views

### 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 ...

**3**

votes

**0**answers

93 views

### 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 $\...

**6**

votes

**2**answers

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 ...

**5**

votes

**1**answer

126 views

### 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 ...

**30**

votes

**6**answers

4k views

### Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a
Lagrangian skeleton ...

**2**

votes

**1**answer

70 views

### 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 ...

**13**

votes

**5**answers

1k views

### Reading list for Equivariant Cohomology

I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...

**6**

votes

**0**answers

349 views

### An algebraic 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 ...

**6**

votes

**1**answer

214 views

### 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.
...

**4**

votes

**1**answer

87 views

### 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 $\...

**0**

votes

**1**answer

123 views

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

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

**0**

votes

**1**answer

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 ...

**10**

votes

**1**answer

321 views

### 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 ...

**15**

votes

**0**answers

239 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 ...

**22**

votes

**4**answers

3k views

### Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...

**1**

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**0**answers

37 views

### 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 $$\...

**8**

votes

**0**answers

206 views

### 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-...

**4**

votes

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110 views

### 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 ...

**3**

votes

**1**answer

390 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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62 views

### 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 ...