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Questions tagged [pseudo-holomorphic-curves]

6
votes
1answer
176 views

Strip breaking phenomenon in the Gromov compactification of Moduli space of Pseudoholomorphic curves

As the title suggests, I want to understand the strip breaking phenomenon that happens when I Gromov-compactify the moduli space of pseudoholomorphic curves from the holomorphic strip $\Bbb R \times [...
1
vote
1answer
160 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 ...
4
votes
0answers
66 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 ...
5
votes
1answer
193 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 ...
4
votes
0answers
106 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 = \...
3
votes
0answers
107 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). ...
9
votes
1answer
395 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 = ...
0
votes
0answers
72 views

Diameter of pseudoholomorphic curves

Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk. My question: can one prove an a-priori bound on the diameter of $u$ (...
4
votes
1answer
217 views

More pseudoholomorphic curves than complex valued functions

A lecture I heard had a remark - "There is a rich class of pseudohoplomorphic curves to a symplectic manifold with an almost complex structure (tamed by the symplectic structure). On the other hand, ...
3
votes
1answer
187 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 ...
11
votes
2answers
317 views

Fredholm property about $L^p$-extension $(p\neq 2)$ of differential operators

The following is a well-known result for elliptic operators. Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact ...
6
votes
1answer
509 views

Deligne Mumford Compactification of Moduli Space Of Annuli

I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" (https://arxiv.org/abs/1001.4593), and it is claimed there, without proof, in section C.4 in the appendix (pp....
4
votes
0answers
72 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
vote
0answers
72 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 ...
7
votes
1answer
350 views

Most general maximum principle for non-integrable almost complex structures

Let $H\subseteq\mathbb C^n$ be a smooth co-oriented real codimension one hypersurface. If $H$ is weakly pseudo-convex, then holomorphic maps $u:\Delta\to\mathbb C^n$ ($\Delta$ denotes the unit disk) ...
4
votes
0answers
65 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 ...
3
votes
0answers
113 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 ...
6
votes
1answer
215 views

In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context. Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...
1
vote
0answers
94 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 ...
14
votes
1answer
425 views

Pseudo-holomorphic curves in the six-sphere

Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...