Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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8 votes
1 answer
780 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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 views

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, ...

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