I have the following exercise:
Let $J$ be an a.c. structure on $U\subset \mathbb{R}^{2n}$ (neighborhood of $0$).
Also, let $f:U \rightarrow \mathbb{C}$ be a function, s.t. $df \circ J=J_{st} \circ df$, where $J_{st}:\mathbb{C} \rightarrow \mathbb{C}, J_{st}(x)=y, J_{st}(y)=-x$.
I want to show that then rank $df_p\neq 1$ and that if $v,w \in \operatorname{ker} df \subset TU$, then $[v,w] \in \operatorname{ker} df$.
For the first part, I have the following idea:
Let $df_p(T_pU)=\operatorname{span}\{v\} \subset \mathbb{C}$. Then $v=a \cdot x + b \cdot y$ where $(x,y)$ are the standard coordinates. Now $J_{st}(v)=ay-bx$ and one can see that $\operatorname{span}\{v\} \neq \operatorname{span}\{ J_{st}(v)\}$. So $df_p(T_pU)$ cant be spanned by 1 vector. Is this correct?
For the second part, I do not really have an idea. I try to use that $J^2=-\operatorname{Id}$. Does anyone have a hint for me?