Questions tagged [symplectic-topology]
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244
questions
3
votes
2
answers
849
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Generating Fukaya category vs split-generating Fukaya category
I just started learning about Fukaya categories and got slightly confused by the following question. It looks like the statement that a collection of objects generate Fukaya category is stronger than ...
3
votes
1
answer
378
views
the tangent space $T_J\mathcal J^k$ of the space of $\omega$-compatible almost complex structure
Let $(M,\omega)$ is a symplectic manifold, and you can assume it is compact if necessary.
Denote by $\mathcal J^k = \mathcal J^k(M, \omega)$ the set of all $\omega$-compatible almost complex ...
4
votes
0
answers
190
views
infinite-dimensional transversality theorem and its application on the universal moduli space of pseudo-holomorphic curves
We would like to discuss Proposition 3.2.1 in McDuff&Salamon's book "J-holomorphic Curves and Symplectic Topology(Second Endition)".
Let me first remind you some background. Let $\Sigma$ be a ...
2
votes
1
answer
343
views
Gromov compactness theorem for genus $g >0$ Riemann surfaces
In McDuff & Salamon's book "J-holomorphic Curves and Symplectic Topology", only the Gromov compactness for spheres has been proved, and I wonder whether or not this compactness result is still ...
7
votes
1
answer
1k
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Definition of "Lagrangian skeleton"
I'm looking for the precise definition of "Lagrangian skeleton", as I'm eventually going to give a talk on this topic. As I asked a professor in my university about references on Lagrangian skeleta, ...
1
vote
0
answers
110
views
Regular almost complex structures on symplectic toric manifolds
Under which assumptions the almost complex structure J defined on a symplectic toric manifold is Fredholm regular for every J-holomorphic sphere?
1
vote
1
answer
171
views
Moser's argument for loops
Let $(\omega_t)_{t\in [0,1]}$ be a path of cohomologous symplectic forms on $X$. The standard Moser's argument shows that there exists a family of diffeomorphisms $(\psi_t)_{t\in [0,1]}$ of $X$ with $\...
3
votes
0
answers
59
views
Connectedness of the space of symplectic embeddings into a higher dimensional manifold
Suppose $M$ and $N$ are symplectic manifolds, $N$ is compact, $\dim_{\mathbb R}(N) \leq \dim_{\mathbb R}(M) -4$. Suppose there are embeddings $f_i:N \to M$, $i=0,1$ such that $f_i^*\omega_M$ is non-...
2
votes
1
answer
283
views
Hofer-Zehnder capacity of toric varieties
Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset \...
5
votes
0
answers
342
views
Grading in Lagrangian Floer homology
What are the conditions on a symplectic manifold (M,w) and on a Lagrangian submanifold L so that Lagrangian Floer complex CF(L, f(L)) is Z-graded? Here f is a compactly supported Hamiltonian isotopy. ...
6
votes
1
answer
503
views
How to understand Taubes' moduli space of holomorphic curves?
Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...
4
votes
0
answers
254
views
Matsushita theorem on framed variety (X,D)
I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact K\...
5
votes
1
answer
233
views
Perburb the Monodromy of Lefschetz fibration over a disk
Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
7
votes
1
answer
370
views
Example where Calabi invariant is nontrivial?
Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of $...
0
votes
0
answers
181
views
Triangulation of S^2xS^2
Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ?
Thanks,
1
vote
0
answers
127
views
Open Hamiltonian Gromov-Witten Invariants
Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants before....
1
vote
0
answers
133
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Shape of the bubbling limit of holomorphic discs
I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...
2
votes
0
answers
183
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holomorphic curves in almost toric fibration and their relation to tropical curves
My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
6
votes
0
answers
423
views
Handle attachment in symplectic category
It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...
3
votes
0
answers
771
views
What is the formula for the homology class represented by the diagonal?
Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...
11
votes
1
answer
1k
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Current status of the linearity of mapping class group
In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
10
votes
1
answer
1k
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Maslov class as a relative cohomology class in $H^2(M, L)$
Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to ...
0
votes
1
answer
322
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Steps in paper on sympl. geometry unclear
I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.
It deals with the question how the stability properties of a ...
4
votes
1
answer
286
views
Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk
Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a ...
4
votes
2
answers
491
views
Legendrian knot in 3-sphere
We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...
9
votes
1
answer
370
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Homology class of a Lagrangian Klein bottle
Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...
3
votes
0
answers
95
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Flux group of surfaces with genus $g\ge2$
I have read in this paper by C. Campos-Apanco and Pedroza (at the end of page 2) that:
When $(M,\omega)$ is a closed surface of genus greater than one, the flux group is trivial.
Does anyone have ...
8
votes
1
answer
805
views
Symplectic reversing diffeomorphisms on a compact symplectic manifold
I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there always a ...
1
vote
0
answers
86
views
Perturbation of a Fredholm sections which preserves compactness of 0-set
I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...
0
votes
1
answer
287
views
Lagrangian flow preserves symplectic form
Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...
0
votes
0
answers
124
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Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?
Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...
4
votes
0
answers
229
views
Generation of compact Lagrangians over fields with characteristic 2
Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...
2
votes
0
answers
289
views
McDuff's classification of symplectic manifolds
According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...
3
votes
0
answers
184
views
Is there an action functional for the s-dependent Floer equation?
The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional $\...
3
votes
2
answers
723
views
Show that the symplectic action 1-form on loop space is closed
I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 \omega(\dot{x}(t)-X_H,Y)...
3
votes
1
answer
454
views
On Gromov's Theorem on Symplectic Homotopy
I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
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?
1
vote
0
answers
80
views
why is there such a 1-form on a planar open book?
Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform page-...
1
vote
0
answers
231
views
Lagrangian fibrations with isolated singular fibers
Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold ...
3
votes
0
answers
252
views
Fully faithful embedding of the exact Fukaya category
Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...
2
votes
0
answers
168
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Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$
I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...
1
vote
0
answers
168
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Does some square of the first Chern class preserved by conifold transition?
Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...
1
vote
2
answers
755
views
When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?
My goal is to better understand the Maslov-index of pseudoholomorphic disks.
For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic ...
1
vote
0
answers
192
views
Connectedness of the symplectic automorphism of the 2-sphere $S^2$
The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is:
Is the group of symplectic automorphisms
of $S^2$ with respect to this ...
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, ...
4
votes
0
answers
154
views
Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres
Given a symplectic 6-manifold $(M,\omega)$ and an embedded
symplectic 2-sphere $C\subset M$ whose normal bundle has the first
Chern class -2. How to find on $M$ another closed 2-form $\eta$ which
only ...
1
vote
0
answers
87
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Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?
Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$
acts holomorphically such that the group $K$ preserves the
Kähler ...
1
vote
0
answers
132
views
Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres
Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...
3
votes
0
answers
261
views
The relative h-principle and extension problems
As a beginner for h-principles, I want to know why the relative
h-principle cannot imply a positive solution to the problems for
extending symplectic structures.
The following is a relative h-...
6
votes
2
answers
1k
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Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory)
I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/...