Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.**Question** Show the flow of $JX$ is by biholomorphisms.


I know one reference to show the answer is yes (where it is stated that: flow of $X$ is by biholomorphisms $\iff$ $X-iJX$ is a holomorphic vector field) but I am interested to find a simple explanation (if indeed one exists).


I would also be interested if anyone knows about the corresponding question for an almost complex manifold $(M,J)$ with a vector field with flow by J-isomorphism's? - for this question I do not have a reference (or clue if it is true).



Vector field with holomorphic flow
A vector field $X$ has flow preserving $J$ just when $L\_X J=0$, i.e. just when $[X,JY]=J[X,Y]$ for any vector field $Y$. Take $Y$ to also have such flow. Many such $Y$ exist locally, spanning the tangent bundle, because $(M,J)$ is a complex manifold. Vanishing of the Nijenhuis tensor gives $[JX,JY]=[X,Y]+J([JX,Y]+[X,JY])$. But then we apply the flows of $X$ and $Y$: $[JX,JY]=[X,Y]+J(J[X,Y]+J[X,Y])=-[X,Y]=J^2[X,Y]=J[X,JY]=J[JX,Y]$. So $[JX,JY]=J[JX,Y]$. So $L\_{JX} J=0$.


