TITLE: Vector field with holomorphic flow
QUESTION [11 upvotes]: 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).

REPLY [2 votes]: Without lose of generality we may work locally in $\mathbb{R}^{2n}$ with its standard complex structure $J$. Let $\phi$ be the flow of the vector field $x'= f(x)$. So the statement in the question is a consequence of the variational equation $$ \partial_t \partial_x \phi_t = Df(\phi). \partial_x \phi_t$$
If the flow is holomorphic then $\partial_x \phi_t$ and its time derivative $\partial_t \partial _x \phi_t $ commute with $J$. This implies that $Df.J= J.Df$. Q.E.D
In the same manner one can shows the following:

Fact: If the flow of a vector field $x'=f(x)$ on $\mathbb{R}^n$ is harmonic then $f$ is  harmonic, too.

Proof:
The Laplacian, the gradient  and the Hessian of a vector valued function is defined naturally by componentwise corresponding operators. Hence the Hessian of a vector valued function is a $3$ dimensional matrix  whose trace is a "vector".
With such notation we have the following formula:
$$\partial_t  \Delta \phi= trace( {(\partial _x \phi_t)}^{tr}Hess(f)\partial_x \phi_t)+ \Delta \phi . \nabla f$$
This shows (using the time independence of $f$ to evaluate at $t = 0$) that if the flow of a vector field is harmonic then the vector field is a harmonic  vector  valued function.
But  what about the  converse? Is the  flow  of  a   harmonic  vector  field, a  harmonic  function? Moreover, how  can we  rephrase the  above  Euclidean fact in an  abstract  Riemannian  manifold?