All Questions
Tagged with pseudo-holomorphic-curves symplectic-topology
22 questions
2
votes
1
answer
194
views
Is a simple J-holomorphic curve injective everywhere except for finitely many points?
Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set
$$
Z:=\{z\...
- 17.5k
2
votes
0
answers
82
views
Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
- 285
2
votes
1
answer
114
views
Question on Gromov-Witten invariants when $A=0$
I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's ...
- 63.9k
2
votes
0
answers
180
views
Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?
Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...
- 21
4
votes
2
answers
448
views
Dismissing pseudoholomorphic curves in embedded contact homology
In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...
- 12.9k
2
votes
0
answers
115
views
Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1
Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
- 113
2
votes
0
answers
42
views
Is the forgetful map a submersion away from the nodal points?
Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow ...
- 21
3
votes
0
answers
102
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 ...
- 461
4
votes
0
answers
193
views
Unobstructedness of nodal holomorphic curve in symplectic manifold
Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...
- 1,761
13
votes
1
answer
506
views
How not to use J-holomorphic curves [closed]
The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
3
votes
1
answer
253
views
Compactly supported symplectomorphisms of $D^2$
I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.
Consider the dimensional disk $D^2 \...
- 2,479
4
votes
1
answer
383
views
Every symplectic submanifold is J-holomorphic
I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am ...
- 1,761
1
vote
0
answers
74
views
Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle
I am trying to understand the construction of symplectic inflation and I am stuck in the following point.
Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \...
- 755
1
vote
1
answer
394
views
Floer equation and Cauchy Riemann equation
Consider a symplectic manifold $(M,\omega)$ with the property that $\pi_2(M) = 0$. Given a time dependent hamiltonian $H_t$ on $M$, and a $\omega$-compatible almost complex structure J on M, we may ...
- 1,242
4
votes
0
answers
110
views
Pairs of J-holomorphic curves
Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex ...
- 755
6
votes
1
answer
741
views
Intuition about bubbling off a ghost bubble
I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that ...
- 17.5k
4
votes
0
answers
127
views
Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$
I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \...
- 755
9
votes
1
answer
740
views
DGLA controlling deformation of holomorphic curves
Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...
- 1,088
4
votes
1
answer
334
views
How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?
When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...
3
votes
0
answers
96
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 ...
- 1,893
4
votes
0
answers
199
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 ...
- 2,789
1
vote
0
answers
147
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 ...
- 1,893