Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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11
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2answers
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
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1answer
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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0answers
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
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3answers
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
1answer
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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0answers
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
2answers
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
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0answers
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
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0answers
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
3answers
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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0answers
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
1answer
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
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0answers
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
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1answer
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
1answer
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
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0answers
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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0answers
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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0answers
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
2answers
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
3answers
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
2answers
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
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0answers
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
2answers
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
votes
0answers
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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0answers
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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0answers
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
1answer
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
1answer
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
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0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
6answers
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
1answer
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
5answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
4answers
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
vote
0answers
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
0answers
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
0answers
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
1answer
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 ...
1
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0answers
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 ...

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