# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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

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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### 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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### 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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### 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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### 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 ...
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### Two possible meanings of “totally real” submanifold

It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent ...
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### On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far. Let me fix some definitions first. ...
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### An example in symplectic geometry

$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...
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### Question about an example in symplectic geometry

Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
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### What is the current status of derived differential geometry?

I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...
In context of Abelian varieties there are a couple of equivalent ways to introduce the polarization of a algebraic variety. One way is to choose a line bundle $\mathcal{L}$ which satisfies certain ...