Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
questions
14
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0
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436
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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 ...
7
votes
1
answer
445
views
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 ...
3
votes
1
answer
1k
views
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 ...
10
votes
0
answers
313
views
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-...
11
votes
0
answers
597
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 ...
2
votes
0
answers
242
views
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 \...
16
votes
1
answer
2k
views
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 ...
1
vote
0
answers
156
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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 $H(t=0)=H_0$ ($...
2
votes
1
answer
107
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
$$
\omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1})
$$
...
5
votes
1
answer
372
views
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.
...
2
votes
0
answers
94
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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-...
1
vote
1
answer
155
views
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 ...
5
votes
0
answers
93
views
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 ...
3
votes
0
answers
203
views
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,...
6
votes
0
answers
170
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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{...
4
votes
1
answer
177
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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 ...
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 ...
3
votes
0
answers
141
views
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 \...
3
votes
0
answers
400
views
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 ...
3
votes
0
answers
91
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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, ...
7
votes
0
answers
214
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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 ...
3
votes
0
answers
433
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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 ...
1
vote
0
answers
518
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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.
...
5
votes
1
answer
557
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 $...
3
votes
0
answers
234
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 $L\subset M$, ...
1
vote
0
answers
79
views
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{...
3
votes
1
answer
320
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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 ...
7
votes
1
answer
273
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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, ...
1
vote
0
answers
191
views
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 $(...
5
votes
1
answer
156
views
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 \...
53
votes
0
answers
2k
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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 ...
2
votes
0
answers
487
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 ...
18
votes
3
answers
3k
views
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 ...
11
votes
0
answers
222
views
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 ...
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 ...
4
votes
1
answer
224
views
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 ...
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 ...
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 ...
4
votes
0
answers
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 ...
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 ...
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 ...
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 $G$ ...
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 $2$), ...
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^{\...
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 $k\geq 1$. ...
10
votes
0
answers
356
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:
...
2
votes
0
answers
230
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 ...
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 $\nu:M\to N$ ...
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 $H^2(M, \mathbb{R})$?
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 $\phi:M\to N$...