How not to use J-holomorphic curves [closed]

Question

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 can avoid committing the sins of their elders, it would be desirable to make a collection of such traps: how not to use holomorphic curves. For instance, the failure of somewhere-injectivity to hold for $J$-holomorphic curves with boundary; the failure of the reparametrization action to be differentiable on Sobolev spaces; inconsistency of gluing with different models of strip-like ends etc.

Of course, there is no need to name names.

If it wasn't obvious, I am looking for: superficially convincing arguments or statements concerning $J$-holomorphic curves that have been used (or almost used) in papers in symplectic topology, and which, on closer inspection, fail to hold for somewhat subtle reasons. I think the answer by Jonny Evans is a great example.

Answer 1

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 branch point where you had your tangency condition then this can fail. As a PhD student, I once used this to prove that all complex curves in $CP^2$ are diffeomorphic to the sphere (fortunately I realised in time that this remarkable result was not correct so it never made it off my blackboard).