Global vector fields

Question

Given a Stein manifold $Y$, I can consider a local vector field around a point $p\in K$, where $K\subset Y$ is compact; e.g. in local coordinates, it can be a constant vector field. In local coordinates I can write $V(z)=v$, on $U$ a neighborhood of $p$. This $V$ can be obviously extended to the whole $\Bbb C^n$: this is because tangent spaces in $\Bbb C^n$ are basically just $\Bbb C^n$, while in a manifold $T_pK$ changes as $p$ changes. So in $\Bbb C^n$ I would get a global flow.

My problem is that this doesn't hold true in a manifold anymore: I would only have a local flow near $p$. I suspect the flow can be holomorphically extended on a whole neighborhood of $K$, but I cannot find any argument for this. Any hint? Thanks

Answer 1

If the manifold $Y$ admits a global (meromorphic) vector field $W$, then around a regular point $p$ (that is, $W(p)\notin \{0,\infty\}$) one has a local holomorphic rectifying chart as you point out. Hence both $W$ and the starting vector field are locally conjugate near $p$. Therefore your question reduces to knowing whether $Y$ admits at least a global meromorphic vector field, but that is well-known (as hinted at in the Wikipedia page https://en.wikipedia.org/wiki/Stein_manifold, see the «spreadable» property).