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        Questions tagged [sg.symplectic-geometry]

        Hamiltonian systems, symplectic flows, classical integrable systems

        1
        vote
        0answers
        69 views

        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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        vote
        0answers
        118 views
        +100

        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 ...
        2
        votes
        0answers
        74 views

        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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        votes
        2answers
        296 views

        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, \...
        2
        votes
        1answer
        123 views

        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-...
        3
        votes
        1answer
        155 views

        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 ...
        1
        vote
        0answers
        112 views

        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 ...
        4
        votes
        1answer
        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 ...
        10
        votes
        1answer
        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 ...
        4
        votes
        1answer
        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 ...
        2
        votes
        0answers
        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 : ...
        7
        votes
        1answer
        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 ...
        8
        votes
        2answers
        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 ...
        8
        votes
        0answers
        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 \...
        2
        votes
        1answer
        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 ...

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