Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
questions
8
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1
answer
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How to construct the mirror partner of a blowup?
Question: Let's assume we have a pair $(X,\check{X})$ that are mirror dual to each other in the sense of Homological mirror symmetry (EDIT: this does not have to be CY n-folds, but can also be a Fano ...
3
votes
1
answer
362
views
Set of singular points for momentum map (with coisotropic action)
Let $G$ be a Lie-group acting on a connected symplectic manifold $M'$ in a hamiltonian way, with an $\operatorname{Ad}^*_G$-equivariant momentum map. Assuming $G$ acts properly on $M'$, we can ...
7
votes
1
answer
658
views
Symplectic form/Kahler metric on a toric manifold
I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form $...
5
votes
0
answers
528
views
Darboux's Theorem
I read proofs of Darboux's theorem that a symplectic form $\omega$ on M locally is symplectomorphic to a standard symplectic form $\omega_{0}= \sum dp \wedge dq$ in Rolf Berndt's An Introduction to ...
18
votes
1
answer
1k
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Categorification of Floer homology
Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a ...
4
votes
0
answers
319
views
Higher genus Cohen-Jones-Segal's conjecture?
Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism
$$(ev,\Phi):\...
3
votes
1
answer
188
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Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor
Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
3
votes
0
answers
149
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Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
11
votes
1
answer
865
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Monopole Floer Homology vs. Heegaard-Floer theory
I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
Is there some version of Atiyah-Floer conjecture ...
1
vote
0
answers
134
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Constructing embedded families of curves with general moduli
Is there a known way to construct flat families of smooth curves $\mathcal{C}/\mathcal{B}$ which are fiberwise embedded in a family of projective varieties $\mathcal{X}/\mathcal{B}$ and which have ...
2
votes
0
answers
134
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What is the difference between the Fukaya category and the Fukaya-Floer category?
What is the difference between the Fukaya category and the Fukaya-Floer category? It seems that the former is defined in terms of flat $U(1)$ line bundles and the latter in terms of flat unitary ...
5
votes
0
answers
414
views
What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
3
votes
1
answer
194
views
Why are the toric fibers of a toric manifold Lagrangian submanifolds?
How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general ...
3
votes
0
answers
265
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Classical analogue of the theorem of equivalence of the S-matrix
In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
8
votes
3
answers
2k
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Conformal-symplectic geometry ?
I think in priciple it's possible to consider a theory of "conformal-symplectic manifolds", in an analogous fashion as the usual conformal geometry.
To spell out the spontaneous definitions: say ...
5
votes
1
answer
334
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
3
votes
1
answer
188
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Index of linearized operator for symplectic vortex equations
In reference [1], the index of the linearized operator for the symplectic vortex equations is computed on page 27-28.
The first step of the proof says that the operator
\begin{equation}\tag{1}
\...
3
votes
1
answer
344
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Is it difficult or easy to find non-symplectomorphic symplectic forms on a manifold?
If people think this question is a little basic I will move it to stackexchange, but I decided to try here first. So it is clear that if two symplectic forms are not cohomologous then they cannot be ...
0
votes
2
answers
500
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
3
votes
0
answers
94
views
Can one choose a sufficiently generic path of a.c.s such that only "codimension 1" bubbling occurs?
Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...
10
votes
3
answers
844
views
Is the 'massive' Calogero-Moser system still integrable?
Background
The (rational) Calogero-Moser system is the dynamical system which describes the evolution of $n$ particles on the line $\mathbb{C}$ which repel each other with force proportional to the ...
2
votes
0
answers
169
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Question on Hori, Iqbal and Vafa's 'D-branes and Mirror Symmetry'
In the paper mentioned above, on page 19, the physics of A-type supersymmetry is related to a Lagrangian submanifold $\gamma$ of a Kaehler manifold $X$. In particular, the phrase "...holomorphic ...
5
votes
0
answers
997
views
Complex but not Symplectic
For every even integer $n>2$, does there exist a smooth $n$ dimensional manifold $M$ that admits a complex structure but not a symplectic one?
2
votes
1
answer
168
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Extensions to symplectomorphisms
Let $G$ be a diffeomorphism of $\mathbb{R}$. Then the map $\mathbb{R}\times\{0\}\rightarrow\mathbb{R}\times\{0\}$ given by
$(x,0)\mapsto (G(x),0)$ may be extended to a symplectomorphism
$(x,\xi)\...
7
votes
1
answer
257
views
Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims
I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...
4
votes
0
answers
233
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Lagrangian submanifold of Poisson manifolds
Let $V$ be a finite dimensional vector space.
Let $\psi\in \Lambda^2V$ be a (possibly degenerate) $2$-vector. Then $\psi$ defines a map $V^*\rightarrow V$. Let $U\subset V$ denote the image of this ...
4
votes
1
answer
511
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The existence of adjoint operator for Sobolev spaces $W^{k,p}(S^2, \mathbb R^n)$
It is known that if $D:H_1 \to H_2$ is a bounded operator between Hilbert spaces, then there exists an adjoint operator $D^* : H_2 \to H_1$ (the field is just $\mathbb R$ rather than $\mathbb C$, so ...
6
votes
0
answers
314
views
Freed-Hopkins-Lurie-Teleman topological boundary conditions v.s. Lagrangian subspace
This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.
For example, we can consider defining the TQFT on a $D^3$ ball with a topological ...
1
vote
0
answers
93
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A genericity argument on family of disconnected holomorphic curves
Let $(W, \lambda)$ be an exact cobordism from $(M_+, \lambda_+)$ to $(M_-, \lambda_-)$ and $\overline{W}$ be the usual symplectic completion of $W$. Let $\mathcal{M}(J)$ be the module space of all ...
1
vote
0
answers
228
views
De Jonquières formula vs. Relative GW invariants
Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety
of linear series of type $\mathfrak{g}^r_d$, i.e.
$$ ...
34
votes
6
answers
10k
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Roadmap for Mirror Symmetry
I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side.
I know basic Complex ...
6
votes
1
answer
378
views
A symplectic form on a symplectic vector bundle
Suppose $E \to B$ is a symplectic vector bundle, i.e. it possesses a fibrewise linear symplectic form $\omega_F$. Further, suppose $\omega_B$ is a symplectic form on $B$.
Question: is there a ...
15
votes
4
answers
1k
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Application of toric varieties for problems that do not mention them
I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?
To give ...
2
votes
0
answers
128
views
Do involutions always stabilize some transverse lagrangians?
Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the ...
5
votes
0
answers
163
views
Virasoro constraints for parametrized GW invariants
Gromov-Witten invariants count isolated stable maps from Riemann surfaces to a fixed symplectic manifold $(M,\omega)$ subject to some incidence conditions. If we instead replace the target manifold ...
4
votes
0
answers
243
views
H-principle for smoothing
I'm trying to find algebraic embedding of singular curves into projective varieties that can be smoothed symplectically but not algebraically.
It's not hard (e.g. using the methods in Hartshorne-...
7
votes
0
answers
227
views
Higher homotopy of diffeomorphism groups from singularities
In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth ...
10
votes
1
answer
591
views
Almost complex structures on $\mathbb CP^2$ that are not tamed
Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...
5
votes
1
answer
246
views
Almost complex structures on a 4-ball that are not tamed
Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $...
7
votes
0
answers
295
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Automorphism that the Fukaya category is "blind" to
Given a symplectic manifold $(M,\omega)$, there is a natural map
$$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$
which sends a symplectic automorphism to the $A_\infty$-functor it induces on the ...
15
votes
0
answers
578
views
Open conjectures on the Fukaya category coming from physics
This is a slightly vague question (for which I apologize in advance): can somebody give examples to open conjectures on the behavior of the $Fuk(M,\omega)$ that come from string theory and can be ...
2
votes
0
answers
159
views
Existence of compatible almost complex structure of symplectic fibration with nice property
Let $\pi: E \to B$ be a fibration over a closed surface $B$ with fier $g(F) \ge 2$. Suppose that both of $B$ and $F$ are closed surfaces and $g(B) \ge2$ and $g(F) \ge 2$. Fix a Kahler structure $(...
6
votes
1
answer
936
views
Does there exists a Fukaya category with no objects
... and really without even the possibility of having objects, so it's not a matter of just finding the "correct" flavour of Fukaya category to use.
Question: Does there exist interesting symplectic ...
7
votes
2
answers
403
views
Infinite dimensional version of a simple fact on certain singular matrices
We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds;
Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with ...
3
votes
1
answer
268
views
Degenerate Reeb orbits
I am reading about contact homology and ECH, and realized that I do not see what goes wrong with the definition of these theories, if one takes into count degenerate Reeb orbits. In general, I would ...
1
vote
0
answers
230
views
Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?
In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...
2
votes
1
answer
271
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An example of Guillemin Sternberg Conjecture
Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^*$ is regular at $0$, ...
5
votes
0
answers
534
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connectedness of moduli space of Calabi-Yau 3-folds by symplectic surgery theory
"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory.
Miles Reid’s Fantasy:“There is only one Calabi-Yau space”
i.e "All CY connected through conifold ...
4
votes
0
answers
90
views
Topology of a convergent sequence of stable maps on a symplectic manifold
Let $(M,\omega)$ be a compact symplectic manifold. Let $J$ be a compatible almost complex structure. Let $g$ be the Riemannian metric corresponding to $\omega,J$.
Let $f_\nu\colon C_\nu\to M$ be a ...
12
votes
6
answers
3k
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When do you go hunting for Lagrangian submanifolds?
Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...