a complex vector field

Question

let $\Omega$ be a closed complex 1-form and $X$ be a complex vector field for which $\Omega(X)=1$ and $\overline{\Omega}(X)=0$.

Why the real and imaginary part of $X$ are pointwise linearly independent and preserve $\Omega$ and $\overline{\Omega}$?

Answer 1

The linear independence is clear, since $\text{Re}(\Omega)(\text{Re}(X)) = 1/2$ and $\text{Re}(\Omega)(\text{Im}(X)) = 0$. Thus, unless $\text{Im}(X)$ is identically $0$, $\text{Im}(X)$ cannot be a scalar multiple of $\text{Re}(X)$. But that eventuality cannot occur, since $\text{Im}(\Omega)(\text{Im}(X)) = -1/2$.

Yes, if $\Omega$ is closed, $\mathscr L_{\text{Re}(X)}\Omega = d\left(\Omega(\text{Re}(X))\right) = d(\frac12) = 0$, and similarly for $\text{Im}(X)$ and $\bar\Omega$.

The remark is that, although there's no reason that $[\text{Re}(X),\text{Im}(X)]$ must vanish, the prototypical situation still seems to be that of $\Omega = dz$ and $X=\partial/\partial z$.