Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi\_{t} : M\rightarrow M $ be its flow. Question: Is $\varphi\_{t}$ a biholomorphism? It is a diffeomorphism but is it holomorphic?



flow of holomorphic vector field
The question is local, so assume that $M$ is open in $\mathbb C^n$, so that we do not have to deal with the second tangent bundle. Then:
\begin{align\*}
\partial\_t \phi\_t &= X\circ \phi\_t
\\
T(\partial\_t \phi\_t) &= \partial\_t T(\phi\_t) = TX\circ T(\phi\_t)
\\
\partial\_t J\circ T(\phi\_t) &= J\circ\partial\_t T(\phi\_t) = J\circ TX\circ T(\phi\_t) = TX\circ J\circ T(\phi\_t)
\\
\partial\_t T(\phi\_t)\circ J &= TX\circ T(\phi\_t)\circ J
\end{align\*}
so $J\circ T(\phi\_t)$ and $T(\phi\_t)\circ J$ solve the same equation with the same initial value $\phi\_0= Id$, thus they are equal. 


