# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,068 questions

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### The norm squared of a moment map

I am studying the paper by E. Lerman:
https://arxiv.org/abs/math/0410568
Let $(M,\sigma)$ be a connected symplectic manifold with a Hamiltonian action of a compact Lie group $G$, so that there exist ...

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### Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)

This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question.
Background
I refer to the following concepts:
Liouville ...

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### Properness of moment map

Suppose that a torus $T$ acts on a non-compact symplectic manifold $M$. Assume that this action is Hamiltonian and that the fixed point set of $T$ is compact. Let $\mu:M\to\mathfrak{t}^{*}$ denote the ...

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### Practical example of Hamiltonian reduction

I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...

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### Metric on the phase space

I am studying PDEs whose symbols satisfy
\begin{equation}
|\partial^\alpha_\xi\partial^\beta_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|}
\end{equation}
for all multi-...

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### Lagrangian surgery

Assume that $L_1$ and $L_2$ are connected Lagrangian submanifolds (of dimension at least 2) which intersect transversally. Do we always get a connected Lagrangian after performing Lagrangian surgeries ...

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### Counting fixed points for Hamiltonian symplectomorphisms on $T^{2}$

This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in ...

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166 views

### Casson invariant and Euler characteristic

A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...

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344 views

### Example of two exotic closed 4-manifolds s.t. SW(X)=0

I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...

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**1**answer

293 views

### Derived algebraic geometry and virtual fundamental cycles: cotangent complexes

I have been thinking of a way to apply the derived algebraic geometry of Toen-Vezzosi to construct virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. This seems to be the ...

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101 views

### Inferring properties of toric manifolds through Delzant's description

Let $(M,\omega, \mathbb{T})$ be a symplectic toric manifold. It is well-known that the properties of $M$ can be retrieved by looking at the moment polytope $\Delta$ image of the momentum map
$$
\mu : ...

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222 views

### Independence of Duistermaat-Heckman measure

Suppose that a compact K?hler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth ...

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262 views

### Kronheimer's results on ALE spaces as hyperkahler quotients

Background: In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a ...

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169 views

### Orlov equivalence between Fukaya categories

In his famous paper https://arxiv.org/abs/math/0503632, Orlov proves the following theorem (for simplicity, let's just focus on the Calabi-Yau case)
Theorem(Orlov): Suppose that $W: \mathbb{A}^d \...

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86 views

### Every symplectic submanifold is J-holomorphic

I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am ...