I know that for a proper, free and smooth action of a Lie group $G$ on a smooth manifold $M$, the quotient space $M/G$ has a smooth manifold structure. I am interested in additional conditions on the action such that if $M$ has a complex manifold structure, then $M/G$ has a complex manifold structure.
I know that this happens if the action is by holomorphisms and $G$ is discrete. However, for nondiscrete groups the example here shows that it's not enough that the action be by holomorphisms in case $G$ is not discrete.
In retracing the proof for the smooth case, the problem I see is that to construct a chart on the quotient, we use a flat chart for the distribution generated by the orbits of the action, and there is no guarantee that this flat chart is holomorphic. So I think the question I'm asking is equivalent to: when can we construct holomorphic flat charts for the distribution generated by the orbits?
Does this at least happen if the group $G$ is itself a complex manifold and the map $G \times M \rightarrow M$ is a holomorphic map? Is there a reference for this?
As a motivating example, I am thinking of the action on the space $M=\mathbb C^* \times \mathbb C^*$ by componentwise multiplication by the group $G=\{ (e^z, e^{\alpha z}): z \in \mathbb C \}$ for some $\alpha$ with nonzero imaginary part. The quotient is still a complex manifold, but why and what is essential to this situation that makes the quotient be a complex manifold?