Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
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Symplectic structure on a vector bundle
Let $G$ be a Lie group and let $M \rightarrow B$ be a $G$-equivariant vector bundle with typical fiber $E$.
Suppose that $B$ and $E$ are both symplectic manifolds with a $G$-Hamiltonian action. Can we ...
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261
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Computation of the Fredhom index in Floer theory
I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
4
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292
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Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology
I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
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131
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Simplest Liouville Manifold not of Finite Type, or - Liouville Cobordism Structure on Pair of Pants?
I've been trying to produce the simplest possible example of a Liouville manifold which wouldn't be of finite type (a Liouville manifold is said to be of finite type if its skeleton is compact), and ...
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101
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DA structure of a Dehn twist
I am trying to find the DA bordered homology structure of a Dehn twist.In https://arxiv.org/pdf/0810.0687v6.pdf page 255 bottom the authors tabulate the differentials of the right module(A side) of ...
3
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255
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Computing the Fredholm index in Floer theory
In Salamon's notes on Floer homology, it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial_s+J_0\partial_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\...
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Compactness properties in floer homology of cotangent bundles in the non-periodic case
Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ ...
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The Pushforward of the Liouville measure
Let's consider a Hamiltonian action of a torus $T$ on a symplectic manifold $(M, \Omega)$. We denote by $\mu: M \rightarrow \mathfrak{t}^*$ the moment map and by $\Omega_\mathfrak{t}(X) := \Omega + &...
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Existence of solutions (along subset of hypersurface) for a system of PDEs
I had asked a similar question about existence of solutions to a system of linear PDEs, and the answer made me realize that I needed to change things to try to make the construction work. So for the ...
2
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Existence of solution to a system of linear PDEs with boundary conditions
I am looking to find a solution, or even just prove the existence of one, to the following system of linear PDEs. They come up in a construction I am trying to work out in symplectic geometry. Here $(...
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Mirzakhani's length function integration formulas and representation varieties
Mirzakhani develops a method to integrate geodesic length functions on moduli space by considering circle bundles over moduli space given by level sets of these functions. There are natural circle ...
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Intuition for almost periodic solution and Poincaré recurrence theorem
I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...
6
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2
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409
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Homogeneous symplectic manifolds
I have often heard/read a statement (see, e.g., this MathOverflow question) equivalent to the following:
Let $G$ be a connected Lie group and $(M,\omega)$ a connected and simply-connected symplectic ...
4
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1
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Family of Dolbeault operators on complex vector bundles over $\mathbb{CP}^1$
Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $...
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Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$
Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...
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Bubbling of disks when proving compactness properties in Lagrangian floer cohomology
When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the ...
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Reference request for showing open(resp. closed) string field theory has A-infinity(resp. L-infinity) algebra structure
I've now begun to study about the relationship between open(resp. closed) string field theory and A-infinity(resp. L-infinity) algebra structure.
For the A-infinity case, I'd already heard that the ...
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Using the removal of singularities theorem in $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ with lagrangian boundary conditions
Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^...
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Maslov index equal to $2$ implies that the disk is not multiply covered
In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ the authors claim that a map $w:D^2\rightarrow S^2$ with $w|_{\partial D^2}\subset L$, where $L$ ...
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Coordinates for quasiperiodic motion after reconstruction
Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
2
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Linearization of $\bar \partial_J$ in the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2
Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for ...
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141
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Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$
In the paper Floer cohomology of Lagrangian intersections
and pseudo-holomorphic disks 2, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $G$ of $\...
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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)$ ...
2
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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$...
2
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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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240
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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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0
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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 ...
2
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1
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339
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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 ...
7
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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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154
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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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257
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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 ...
2
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1
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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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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 ...
6
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2
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930
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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 ...
3
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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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0
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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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319
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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 ...
3
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111
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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 $\...
5
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270
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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 ...
7
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2
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455
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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 ...
4
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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)$$ where $g(\alpha)$ is the genus of a pseudoholomorphic representative of the Poincaré dual of $\...
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1
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156
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Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
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A theorem about the symplectic geometry of projective bundles
I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta ...
0
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1
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210
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Hamilton equations-Symplectic scheme [closed]
We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
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0
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Maslov cycle for the Conley-Zehnder index - what are its regular points?
I'm looking at the definition of the Conley-Zehnder index, where it is important to look at the group $$\text{Sp}(2n)^* := \{ A \in \text{Sp}(2n) | \det (A-\text{Id}) \neq 0 \}$$and its complement $$\...
2
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1
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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 ...