Questions tagged [symplectic-topology]

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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 ...
Juan Gonzalo's user avatar
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 ...
Hang's user avatar
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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 ...
Hang's user avatar
  • 2,719
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 ...
Hang's user avatar
  • 2,719
7 votes
1 answer
1k views

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, ...
Math.StackExchange's user avatar
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?
alephx's user avatar
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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 $\...
Mohammad Farajzadeh-Tehrani's user avatar
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-...
Anon's user avatar
  • 768
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 \...
Kiu's user avatar
  • 883
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. ...
John's user avatar
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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 ...
trick1234's user avatar
  • 155
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\...
user avatar
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 ...
trick1234's user avatar
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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 $...
user avatar
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,
SKShukla's user avatar
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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....
Meer Ashwinkumar's user avatar
1 vote
0 answers
133 views

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 ...
Yaniv Ganor's user avatar
  • 1,873
2 votes
0 answers
183 views

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 ...
Sofie's user avatar
  • 21
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 ...
jhgfd's user avatar
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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 $M\times M$. ...
Ritwik's user avatar
  • 3,235
11 votes
1 answer
1k views

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 ...
user6419's user avatar
  • 431
10 votes
1 answer
1k views

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 ...
Momchil Konstantinov's user avatar
0 votes
1 answer
322 views

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 ...
Zlatan12's user avatar
  • 181
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 ...
Trick1234's user avatar
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 $\...
nikita's user avatar
  • 1,335
9 votes
1 answer
370 views

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 ...
Elizabeth S. Q. Goodman's user avatar
3 votes
0 answers
95 views

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 ...
Daniel Robert-Nicoud's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
MBIS's user avatar
  • 529
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 ...
Physicist 2.0's user avatar
0 votes
0 answers
124 views

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 ...
Ritwik's user avatar
  • 3,235
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 ...
YHBKJ's user avatar
  • 3,127
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 ...
nikita's user avatar
  • 1,335
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 $\...
PeteT's user avatar
  • 61
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)...
PeteT's user avatar
  • 61
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 ...
Tatin's user avatar
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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?
user66943's user avatar
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-...
nikita's user avatar
  • 1,335
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 ...
YHBKJ's user avatar
  • 3,127
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 ...
YHBKJ's user avatar
  • 3,127
2 votes
0 answers
168 views

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 ...
YHBKJ's user avatar
  • 3,127
1 vote
0 answers
168 views

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 ...
Jiang's user avatar
  • 153
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 ...
Dave's user avatar
  • 281
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 ...
Entaou's user avatar
  • 285
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, ...
Jan-David Salchow's user avatar
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 ...
Jiang's user avatar
  • 153
1 vote
0 answers
87 views

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 ...
LTY's user avatar
  • 11
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 ...
aeg's user avatar
  • 31
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-...
user57989's user avatar
6 votes
2 answers
1k views

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/...
Yaniv Ganor's user avatar
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