Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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How should we think about the algebraic moment map?

My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...
Hugh Thomas's user avatar
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Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?

Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space. According to Gromov's h-principle for contact embeddings, any contact manifold ...
Vivek Shende's user avatar
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Arnold's book on classical mechanics [duplicate]

Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other ...
user174848's user avatar
10 votes
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Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold

Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
Mohan Swaminathan's user avatar
11 votes
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598 views

Does quantum cohomology have an $E_\infty$-ring structure?

Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication ...
QuantumRing's user avatar
2 votes
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Fixed point set of diffeomorphism is a submanifold

I am in the following setting: Let $(\Sigma,\alpha\vert_\Sigma)$ be a compact regular energy surface of restricted contact type in an exact Hamiltonian manifold $(M,d\alpha,H)$. Given $\varphi \in \...
TheGeekGreek's user avatar
16 votes
1 answer
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Why is embedded contact homology so powerful?

The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in ...
Shaoyun Bai's user avatar
1 vote
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symplectic Runge-Kutta for matrix differential equation

I would like to solve, for $t>0$ the following matrix differential equation: $$U'(t)=H(t)U(t)$$ with initial condition $U(t=0)=U_0$ ($2N\times2N$, symplectic and unitary matrix) and $H(t=0)=H_0$ ($...
Giuseppe P.'s user avatar
2 votes
1 answer
108 views

$2$-Form inducing a non-degenerate form on $\Gamma(T\mathbb{R}^{2n+1})$

Every $2$-form $\omega\in \Omega^2(\mathbb{R}^{2n+1})$ induces a skew-symmetric map $$ \omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1}) $$ ...
ThorbenK's user avatar
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Stabilizer groups of Yang-Mills connections

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface. ...
Tobias Diez's user avatar
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How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...
no_idea's user avatar
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1 answer
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Deform a complex structure fixing marked points

Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism ...
Math1016's user avatar
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Lagragian floer homology vs homology of $\Omega(L_0,L_1)$

I'm very new to this subject, so apologies for a very naive question and probably many mistakes. Let $M$ be some compact sympletic manifold with $L_0,L_1$ Lagrangian submanifolds which intersects ...
davik's user avatar
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3 votes
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Regularity of the dependence of the flow on the vector field definining it

Let $M$ be a smooth compact manifold and $k \geqslant 1$. Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
Thibaut Mazuir's user avatar
6 votes
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Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$

I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...
cr1t1cal's user avatar
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4 votes
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Is every $M\in \mathfrak{s}\mathfrak{p}_4(F)$ conjugate to an "upper triangular" matrix?

Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the ...
user299843's user avatar
2 votes
0 answers
99 views

Symplectic form on $\Omega^0(X,End(E))$

Let $E\rightarrow X$ be a holomorphic vector bundle over a Kahler manifold. Is there a natural symplectic form on the space $\Omega^0(X,End(E))$ ? For example on $\Omega^1(X,End(E))$ we have the ...
BinAcker's user avatar
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Is composition of discrete Hamiltonian flows integrable?

Consider $\Bbb{R}^2$ with the usual symplectic form $$\omega = dx \wedge dy$$ For a function $H \colon \Bbb{R}^2 \to \Bbb{R}$, let $X_H$ be the Hamiltonian vector field. Then the map $\Bbb{R}^2 \to \...
Nick's user avatar
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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 ...
WRehman's user avatar
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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, ...
Filip's user avatar
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7 votes
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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 ...
Riccardo's user avatar
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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 ...
Quarto Bendir's user avatar
1 vote
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526 views

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. ...
BrianT's user avatar
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5 votes
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558 views

Geometric invariants of a Riemannian manifold encoded in certain moment map

Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...
Ali Taghavi's user avatar
3 votes
0 answers
235 views

Natural equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$. A Lagrangian sphere $L\subset M$, ...
Merlin Christ's user avatar
1 vote
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Symplectic structure on the space of complexes of holomorphic vector bundles

Let $E\rightarrow X$ be a holomorphic vector bundle over a complex manifold. Denote by $Dol(E)$ the space of holomorphic structures on $E$. Fix any Hermitian metric $h$ on $E$ and denote by $\mathcal{...
BinAcker's user avatar
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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 ...
Maria's user avatar
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1 answer
273 views

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, ...
Maria's user avatar
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polarization of symplectic manifold

The geometric quantization can be considered as an approach the formalize the way of associating a quantum theory corresponding to a given classical theory. Suppose we start with a sympetic manifold $(...
user267839's user avatar
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5 votes
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Extension of a holomorphic curve in $B^4$ to one in $\mathbb{C}P^2$

Let $B^4$ be the closed unit ball in $\mathbb{C}^2$ and $J$ an almost complex structure sufficiently closed to the standard complex structure on $\mathbb{C}^2$ in the $C^0$-topology. Let $u \colon S \...
Math1016's user avatar
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53 votes
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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 ...
davik's user avatar
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2 votes
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490 views

Polarizations in algebraic and symplectic geometry

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 ...
user267839's user avatar
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18 votes
3 answers
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Applications of symplectic geometry to classical mechanics

It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space. I am ...
asv's user avatar
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11 votes
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Mathematical pendulum and $\mathbb C P^n$

I am very puzzled by the following remark on p.346 in Arnold's book "Mathematical methods of classical mechanics": Another method of construction the same symplectic structure on complex ...
Nikita Kalinin's user avatar
2 votes
1 answer
232 views

Extension of a holomorphic vector bundle on a nodal curve

I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve. Let $C$ be a nodal curve without closed componets and $E$ a ...
Math1016's user avatar
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4 votes
1 answer
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Embeddings of magnetic cotangent bundles over surfaces into closed symplectic 4-manifolds

Let $\Sigma$ be a closed, orientable surface. Then the cotangent bundle $T^*\Sigma$ has a canonical symplectic form $\omega$, given as the derivative of the tautological Liouville one-form. We can ...
Rohil Prasad's user avatar
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3 votes
0 answers
99 views

Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries

Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
ChiHong Chow's user avatar
4 votes
0 answers
213 views

Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric. I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
gigi's user avatar
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4 votes
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131 views

Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?

let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...
Yuan Yao's user avatar
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3 votes
0 answers
71 views

Holomorphic homeomorphisms

Let $M$ be a connected closed smooth manifold. Consider the group $\mathrm{Homeo}(M)$ of homeomorphisms $M\to M$ endowed with the $C^0$-topology. If $M$ has a symplectic structure some people study ...
user avatar
6 votes
0 answers
490 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
TheWildCat's user avatar
1 vote
1 answer
130 views

Marsden–Weinstein: example of not proper action

In order to apply the Marsden–Weinstein reduction, the action of the group $G$ must be free and proper. On the other hand, if I correctly understand, the M-W reduction obtained from a given group $G$ ...
Doriano Brogioli's user avatar
5 votes
0 answers
93 views

Area preserving diffeomorphisms of surfaces with only hyperbolic periodic points

This is a (probably very naive question) about area-preserving maps of surfaces. Does there exist a Hamiltonian diffeomorphism $$ f: \Sigma \to \Sigma $$ of a symplectic surface (real dimension $2$), ...
skr's user avatar
  • 59
2 votes
1 answer
226 views

Elliptic operators and Leibniz rule

Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule? Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\...
Ali Taghavi's user avatar
2 votes
1 answer
262 views

Smooth covers rescaling the symplectic form

Let $(M, \omega)$ be a connected closed symplectic manifold of dimension $2n$. Assume there exist smooth covering maps $\phi_k:M\to M$ such that $\phi_k^* \omega=\sqrt[n]{k}\omega$ for all $k\geq 1$. ...
user avatar
10 votes
0 answers
361 views

Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck

Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs: ...
Quarto Bendir's user avatar
2 votes
0 answers
231 views

What is the relationship between Riemannian and sympletic musical isomorphisms on the cotangent bundle?

Let $M$ be a smooth manifold. Its cotangent bundle naturally has a symplectic structure, and this gives rise to musical isomorphisms. These musical isomorphisms are the ones from physics that relate ...
Tim Phalange's user avatar
4 votes
0 answers
103 views

Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates

Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$. Assume there is a diffeomorphism $\nu:M\to N$ ...
user avatar
3 votes
1 answer
160 views

At most countably many symplectic forms in given cohomology class

Let $M$ be a connected closed smooth manifold. Are there at most countably many non-diffeomorphic symplectic forms in any given class in $H^2(M, \mathbb{R})$?
user avatar
2 votes
1 answer
189 views

Non-symplectomorphic isometric compact Kähler manifolds

Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$. Assume there is a diffeomorphism $\phi:M\to N$...
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