Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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Generic choice of non-degenerate Hamiltonians $H$ in Floer theory

When developing floer theory for an Hamiltonian $H:M\times S^{1}\rightarrow \mathbb{R}$ we will want $H$ to satisfy a non-degenerancy condition, that is, for every $x\in \mathcal{P}(H)$, a periodic ...
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Positive symplectic homology with different Liouville filling

Assume $(Y, \alpha)$ has an exact Liouville filling, i.e. there exists $(W, d\lambda)$ with $Y=\partial W, \alpha=\lambda|_{Y}$. Then for two different exact Liouville fillings $(W_1,d\lambda_1), (W_2,...
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Are all exact Lagrangian spheres, vanishing cycles?

Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F_\theta = \pi^{-1}(\theta)$ over that ...
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Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"

I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$. ...
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What is the interpretation of Jacobi Identity on sympletic manifold?

Context (pg-321): We have a manifold with an anti symmetric metric tensor/sympletic form $S$ with components in a basis $S_{ab}$ satisfying the property that $$dS=0$$ Where $d$ is the exterior ...
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symplectic gromov witten invariants of weighted projective space

Does anyone know if the symplectic (genus zero) GW invariants have been computed for weighted projective spaces? It seems there are already algebraic computations https://arxiv.org/abs/math/0608481 Is ...
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On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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Existence of non-trivial "line-symplectic" manifolds

One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...
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Linearization of the Floer equation

In Floer Homology we want to prove that the Moduli spaces $\mathcal{M}(x^{-},x^{+})$ are finite dimensional manifolds. This is done by expressing them as the zero set of a Fredholm map. First one ...
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Gromov width of weighted projective planes

I am interested in knowing the Gromov width of (the complement of the three orbifold points of) weighted projective planes $\Bbb{CP}(a,b,c)$. Let me emphasize that I am mainly interested in upper ...
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Does $(S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$ admit a symplectic form?

This is a crosspost from this MSE question from a year ago. Consider the smooth four-manifold $M = (S^1\times S^3)\#(S^1\times S^3)\#(S^2\times S^2)$. Does $M$ admit a symplectic form? If $\omega$ ...
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Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?

Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...
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Gluing of Morse-type trajectories in "Floer Homology of Cotangent bundles"

In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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Gluing of hybrid trajectories in Floer homology

In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
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Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$. Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...
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Symplectic cohomology of $T^* \mathbb{CP}^2$

I'm looking for an explanation for why the symplectic cohomology $SH^*(T^* \mathbb{CP}^2,\mathbb{Z})$ is 2-torsion (I heard this in passing; perhaps it's not even true!). By a clever argument that I ...
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Results proved using the geometry of moduli spaces of pseudo-holomorphic curves?

It is well-known that the Gromov-Witten invariants and their Floer-theoretic counterpart of symplectic manifolds have rich algebraic structures. However, sometimes it's quite useful even by ...
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Chern number for curves in a toric variety

Suppose $X$ is a non-singular toric variety that is the closure of the complex torus $X^\circ \simeq T_{\mathbb C}$. Let $Y_1,\dots, Y_N$ be the closures of codimension one $T_{\mathbb C}$-orbits, so ...
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Motivation behind the usual setting of the (weak) Arnold conjecture for fixed points of an hamiltonian diffeomorphism

I'm trying to find out the motivations that led V. Arnold to formulate his famous conjecture (I guess theorem by now) in the following form: Let $(M,\omega)$ be a closed symplectic manifold (add ...
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Multisymplectic connections and topological invariants

I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry. From symplectic connections, it is well ...
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Charts for the Banach manifold of smooth almost complex structures $\mathcal{J}^{l}$

Consider the closure in the $C^l$-topology of the space of smooth almost complex structrues of a symplectic manifold $(M,\omega)$. We will denote this space by $\mathcal{J}^l$. It's a very used fact ...
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$\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$

Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true ...
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plumbing description of resolution of ADE singularities

Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...
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Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic

Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...
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Sectional curvature and injectivity radius of natural metric in Cotangent bundles

In the following paper by Cielibeak, Ginzburg and Kerman https://arxiv.org/pdf/math/0210468.pdf they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is bounded ...
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Proving that a twisted cotangent bundle is geometrically bounded

In Cieliebak, Ginzburg and Kerman's paper Symplectic homology and periodic orbits near symplectic submanifolds, the authors claim and give a proof that a twisted cotangent bundle will be geometrically ...
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ECH Capacities for general domains

With this post I want to ask for some guidance for writting a program to calculated ECH capacities of toric domains. Below a i give some more explanations of what i am working on. Lets consider a ...
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Fujiki class $\mathcal C$ with a symplectic structure

Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ ...
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Influence of symplectic invariants of the complement on being superheavy

Let $(M,\omega)$ be a symplectic manifold. I'm trying to show that a compact subset $K\subset M$ is $1$-superheavy [1, Definition 1.3] where $1=PD([M])$ is the unit in $QH^0(M)$. My question is: How ...
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If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?

Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
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Cup product and PSS map

Let $(M,\omega)$ be a symplectic manifold and let $H$ a Hamiltonian function. If $M$ is not closed we consider $H$ to be linear at infinity to ensure that $HF^*(H)$ is well-defined (I'm particularly ...
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From Delzant polytope to lattice polytope

By definition, an $n$-dimensional Delzant polytope $P$ is not necessarily a lattice polytope. But is there a natural way (or operations) to turn $P$ into a lattice polytope using the fact that the ...
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Continuation principle for solutions of Floer's equation in $\mathbb{R}\times [0,1]$ and transversality

Consider $(M,\omega)$ a symplectic manifold and $J$ a compatible almost complex structure. For me it's well known that if we consider 2 solutions $u,v:\mathbb{R}\times S^1\rightarrow M$ of Floer's ...
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3 votes
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Symplectic orbits in projective Hilbert spaces are simply connected

Let $G$ be a connected Lie group and let $(\pi, \mathcal{H})$ be an irreducible unitary representation of $G$ on an infinite-dimensional Hilbert space $\mathcal{H}$. Denote by $\mathcal{H}^{\infty}$ ...
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Writting the Floer map in local coordinates using the exponential chart

Following Salamon's Notes in Floer Homology , consider the Floer equation $$\mathcal{F}(u):=\partial_su+J_t(\partial_tu+\nabla H_t(u))=0$$ Then we can write in local coordinates $$\mathcal \Phi_u^{-1}(...
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1 answer
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Proof of the Hamiltonian slice theorem

Theorem(Guillemin Sternberg Marle) Let $(M, \omega, \mu) $ be a symplectic manifold together with a Hamiltonian group action. Let $p$ be a point in $M$ such that $O_p $ is contained in the zero level ...
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Compatible almost complex structures such that the associated riemannian metric has positive injectivity radius

Let $M$ be a compact manifold, consider $\omega$ the canonical symplectic form in $T^*M$ and $\hat J$ the canonical almost complex structure coming from the Sasaki metric. Let $\mathcal{J}$ be the set ...
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5 votes
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Which symplectic manifolds are coadjoint orbits of finite-dimensional $G$ on $\mathfrak{g}^*$ with the Kostant–Kirillov–Souriau Poisson structure?

Given some finite-dimensional symplectic manifold $(M,\omega)$, can we answer whether or not it arises as the coadjoint orbit of some finite-dimensional $G$ acting on the dual $\mathfrak{g}^*$ to its ...
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4 votes
2 answers
271 views

Dismissing pseudoholomorphic curves in embedded contact homology

In the papers The periodic Floer homology of a Dehn twist, Rounding corners of polygons and the embedded contact homology of $T^3$, and Combinatorial embedded contact homology for toric contact ...
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2 votes
1 answer
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Associativity of orientations of determinant bundles in Floer homology

I have been reading the paper "Coherent orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that ...
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Lie derivations of algebra of smooth functions in a symplectic manifold

Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson ...
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Are there connected closed 4-manifolds admitting a regular Almost Lagrangian distribution, and which are not Lorentzian?

In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Almost Lagrangian distribution and ...
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1 vote
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Symplectic structure on moduli space of holomorphic Abelian differentials

I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
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Confusion about notation

I recently came across the following generalization of the Darboux-Weinstein lemma: Let $N$ be a manifold endowed with two symplectic forms $\omega_1, \omega_2$, and let $P$ be a compact submanifold ...
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4 votes
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Are there a geometry behind the singularity formation in solutions to nonlinear ODE's?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic. If we take two apparently simple first order ...
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11 votes
1 answer
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Asking whether there is a compact Lie group containing affine symplectic group

The affine symplectic group is interesting and important in physics. However, the Lie group is noncompact. In order to have some good properties (Basically, we need some good behavior of Haar measure) ...
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9 votes
1 answer
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"Nice" way to compute the signature of a toric manifold?

Is there a "nice" way to compute the signature of a smooth toric manifold of even complex dimension in terms of the moment polytope? By signature I mean in the sense of topology (see https://...
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2 votes
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Compactness of Moduli spaces in Lagrangian Floer Cohomology

I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
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Removal of singularities theorem for J-holomorphic curves in non-compact manifolds

Following the book by Mcduff and Salamon, J-holomorphic curves and Symplectic Topology, we know that every $J-$holomorphic curve on the punctured disk with values in a compact symplectic manifold ...
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1 vote
1 answer
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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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