Let X be a compact symplectic manifold and $H_t,J_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a single marked point at the origin $\mathfrak{o}$ and a positive strip like end at $+\infty$. The maps satisfy Floer's equation for the pair $(\mathbb{J},\mathbb{H})$ below.

We consider a surface dependent complex structure $\mathbb{J}$ which is *domain-independent* (agreeing with some generic J') near the origin $\mathfrak{o}$ and agrees with $J_t$ along the strip-like end.

Similarly, we have a surface dependent $\mathbb{H}$ which is 0 near the origin and which agrees with $H_t$ along the strip-like end.

In proposition 4.1 of their paper on gluing Floer trajectories:

http://arxiv.org/pdf/0711.4187v3.pdf

Oh and Zhu make the following claim which "can be derived by a standard argument."

Transversality for maps as above from $\mathbb{C} \to X$ can be achieved using a generic $\mathbb{J},\mathbb{H}$ of the above form.

This proposition doesn't raise any eyebrows because my general impression is that most of the transversality difficulties come from sphere bubbling in these problems. But I'm wondering exactly how it is proven? I could imagine two slightly different types of arguments. I SHOULD NOTE THAT I'M MOSTLY INTERESTED IN THE FACT THAT FOR A FIXED NONDEGENERATE $\mathbb{H}$, we can perturb $\mathbb{J}$ in the class above to achieve transversality.

One approach would be to use the fact that our *domain is stable* and to draw upon the circle of ideas from Ruan-Tian as explained for example in section 6.7 of MacDuff and Salamon's big textbook on J-holomorphic curves.

The second is to follow the lines of Floer, Hofer and Salamon's paper http://www.math.ethz.ch/~salamon/PREPRINTS/trans.pdf in particular try to prove somewhere injectivity for such solutions as in Theorem 4.3 and then follow the proof of Theorem 5.1.

Can either of these approaches be adapted to the present situation? If not, what is this "standard argument ?" I'm curious just because the Floer datum are *somewhat* constrained in that one isn't allowing *arbitrary* domain-dependent perturbations so I'm wondering how that might enter into the proof.