Questions tagged [pseudo-holomorphic-curves]

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Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold

Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
Mohan Swaminathan's user avatar
6 votes
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Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$

I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...
cr1t1cal's user avatar
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A couple of questions about the moduli space of annuli with some marked points on the boundary components

I'm trying to work out an answer for my previous question and I'm stuck with the following issue: In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
Riccardo's user avatar
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Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology

I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
Riccardo's user avatar
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4 votes
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Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?

let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...
Yuan Yao's user avatar
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4 votes
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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 ...
cr1t1cal's user avatar
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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 = \...
cr1t1cal's user avatar
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4 votes
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186 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). ...
Mohan Swaminathan's user avatar
4 votes
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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 ...
asv's user avatar
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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
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Bubbling off a sphere in a splitting/stretching manifold

This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology ...
Riccardo's user avatar
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3 votes
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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 ...
ChiHong Chow's user avatar
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A clarification on why the injectivity radius is involved in Lemma 10.7 of Compactness results in Symplectic Field Theory by B.-E.-H.-W.-Z

I'm trying to understand why in the following lemma (Lemma 10.7 of [BEHWZ]), the upper bound on the $L^{\infty}$-norm of the differential is given in terms of the injective radius w.r.t to a specific ...
Riccardo's user avatar
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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 ...
Yaniv Ganor's user avatar
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2 votes
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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 ...
ChoMedit's user avatar
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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 ...
OSBM's user avatar
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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 \...
Yuan Yao's user avatar
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2 votes
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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 ...
Amanda's user avatar
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1 vote
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207 views

Holomorphic strips versus holomorphic disks

What are the advantages of using holomorphic strips in Lagrangian intersection theory instead of using holomorphic disks? I understand that for the analysis of strips one often needs to choose ...
Klaus Niederkrüger's user avatar
1 vote
0 answers
87 views

symplectic gromov witten invariants of weighted projective space

Does anyone know if the symplectic (genus zero) GW invariants have been computed for weighted projective spaces? It seems there are already algebraic computations https://arxiv.org/abs/math/0608481 Is ...
Yuan Yao's user avatar
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1 vote
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71 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 \...
cr1t1cal's user avatar
  • 755
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 ...
trick1234's user avatar
  • 155
1 vote
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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
0 votes
0 answers
77 views

Estimate for the diameter of the image of a holomorphic disk by the area of the holomorphic disk

Let $(M, J, g)$ be a compact almost complex manifold with a Riemannian metric $g$ that preserves the almost complex structure $J$. I want to prove that a holomorphic disk $u: D^2\to M$ of a small area ...
Shah Faisal's user avatar