# Tagged Questions

Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

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### Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a vector space together with fixed symplectic bilinear ...
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### Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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### Passage in a proof from Hofer-Zehnder

The proof I'm referring to is to the following theorem. Assume $S$ is a compact regular and strictly convex energy surface for the Hamiltonian field $X_H$ in $\mathbb{R}^{2n}$. Then $S$ carries a ...
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### Additional References — Symplectic Geometry

Please forgive me if I should ask this question somewhere else. If so, let me know and I will do so. I have never asked a question of this nature before, and so am unsure where an appropriate place ...
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### Wedge Products with the Symplectic Form

Let $\omega$ denote the symplectic form on $\mathbb{R}^{2n}$, namely $\omega = dx_1 \wedge dy_1 + \cdots + dx_n \wedge dy_n$. Then let $T$ be the linear map from $(n-1)$- forms to $(n+1)$-...
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### Momentum Map-Submersion

Let $(M,\omega)$ be a symplectic manifold and $G$ a Lie group acting hamiltonian on $M$, such that the momentum map $\Phi \colon M \to \mathfrak{g}^*$ is $G$-equivariant w.r.t. the coadjoint-action on ...
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### $\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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### Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
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### Darboux theorem for symplectic manifold of degree 2

Given $p \in M$ and $\alpha \in \Omega^1(M)$ with $\alpha_p \neq 0$, show that there exists a neighbourhood $U$ of $p$ in $M$ and $f,g \in C^{\infty}(U)$ such that $\alpha|_U = f dg$. To show this ...
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### Specifics on the Williamson normal form algorithm

I'm looking at the algorithm for Williamson normal form for symplectic diagonalization of positive-definite symmetric real matrices, given on pp.24 here: https://www.ime.usp.br/~piccione/Downloads/...
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### Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
I am trying to solve the following exercise: Let $(M,\omega)$ be a symplectic manifold and $L$ a compact Lagrangian submanifold such that $H^{1}(L)=0$. Let $\{L_{t}\}_{t\in(-1,1)}$ be a smooth family ...