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I noticed this question has been bumped up to the front page, and the most recent answer is about 8 years old: the subject has moved on since then, and more has been written. Here is my understanding …

Here are two answers to your first question: Yes your argument works. To see that there are no curves for a generic K3 observe that their homology classes must be Poincare dual to an integral (1,1)- …

I think that YangMills is probably right that Donaldson never wrote the conjecture down. But there are some interesting circles of ideas surrounding the conjecture which deserve mention which, again h …

On the suggestion of DamienC, I'm converting my comments into an answer. I didn't do this before because I don't really know the answer to the deeper question of why the microlocal sheaf category is t …

The Kontsevich recursion formula is not special to $\mathbf{CP}^2$. It is a particular application of the more general associativity formula for quantum cohomology, which is something true for symplec …

If you think instead of the prequantum line bundle (i.e. the complex line bundle associated to your prequantum circle bundle using the standard representation of the circle on $\mathbb{C}$) then the s …

Neck-stretching is a deformation of an almost complex structure in a neighbourhood of a hypersurface. In the Eliashberg-Givental-Hofer paper, neck-stretching is called "splitting along a contact subma …

One classic thing to do is to take a sequence of holomorphic curves with a tangency condition and assume that the tangency condition still holds in the limit: if the limit is a multiple cover with a b …

When you try and prove that $d^2=0$ ($d$ being the Floer differential) you need to look at the boundary of the moduli space of index 2 J-holomorphic strips with one boundary on $L_0$, one on $L_1$. Ce …

Floer homology has, in one form or another, become ubiquitous in symplectic geometry and to give a complete list of its applications would be a mammoth task. Here are a few. 1) One early incarnation …

This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (am …

I'm not sure if this is quite what you're looking for, but I always found it a useful example to keep in mind. Take the genus 1 invariant for degree 1 maps into $P^1$. There are no degree 1 holomorphi …

Extend $u$ to get a $C^1$ pseudoholpmorphic map defined on $\mathbb{C}$ by setting $u$ constant outside the unit disc. It's $C^1$ because you know the derivative of $u$ along the unit circle vanishes …

By the time I finished writing this answer someone has explained the whole idea in a comment, but I thought I'd post it anyway as there is some more detail here. I assume the version of Arnold-Liouvil …

I wholeheartedly agree with both of Chris Gerig's suggestions. The small McDuff-Salamon book on holomorphic curves ("J-holomorphic curves and quantum cohomology", available on McDuff's webpage) has a …