In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried to verify this but it gives me an error. Maybe I understood it wrong? Here is what they wrote: Let $(S,i)$ be a complex surface and $(V,J)$ an almost complex manifold. If $f$ is a differentiable map $S\rightarrow V$, we define $\overline{\partial}f=\frac{1}{2}(df+J(f)dfi)$. We consider is as a section over $\text{graph}(f)$ of the vector bundle $E=\overline{\text{Hom}_{J}}(TS,TV)$ over $S\times V$ (the fiber $E_{z,v}$ is the space of anti-complex morphisms from $T_{z}S$ to $T_{v}V$). This gives a meaning to the equation $\overline{\partial}_{J}f=g$ if $g$ is a global section of $E$. An essential observation is that this equation is equivalent to the fact that the map $\text{Id}\times f:S\rightarrow S\times V$ is $J_{g}$-holomorphic for the structure $J_{g}$ on $S\times V$ defined by $J_{g}(\xi,X)=(i\xi,JX+g\xi)$.
I could check that $J_{g}$ is indeed an almost complex structure (i.e. $J_{g}^{2}=-\text{Id}$). What I do not understand is: Assuming that $\frac{1}{2}(df+J(f)dfi)=g$ why is then $df_{g}+J_{g}df_{g}i=0$ whare $f_{g}=\text{Id}\times f$?
Here is what I calculated: For $\xi \in TS$ we have $df_{g}\xi=(\xi, df\xi)$ and $J_{g}df_{g}i\xi = (-\xi , Jdfi\xi + g(i\xi))$. Adding these up we obtain $df_{g}\xi+J_{g}df_{g}i\xi=(0,2g(\xi)+g(i\xi))$ which is obviously not $0$. Where is the mistake? Did I understood something wrong?
Arthur