Let $G$ be a complex Lie group acting on a complex analytic space $X.$ To be clear, I don't require $X$ to be reduced. Let $f: Y\rightarrow X$ be a smooth morphism such that the $G$-action lifts to $Y.$ By a smooth morphism I mean a flat surjective morphism with smooth fibers: (should I call this faithfully flat)?
Suppose that the $G$-action on $X$ is free and proper, where proper means that the graph of the action $G\times X\rightarrow X \times X$ is a proper map of the underlying topological spaces and free means trivial stabilizers. I know that the hypotheses imply that the orbit space $X/G$ admits the structure of a complex analytic space.
Is the following "theorem" true?
"Theorem:"
$\bullet$ The projection $\pi: X\rightarrow X/G$ is a smooth morphism.
Given my current understanding, this seems true, but I fear I might be misconstruing some subtle point.
$\bullet$ There exists a complex analytic space $\underline{Y}$ and a smooth morphism $\underline{Y}\rightarrow X/G$ such that $\pi^{\star}{\underline{Y}}\simeq Y.$
Here, the pullback $\pi^{\star}$ always denotes the fiber product in the category of complex analytic spaces.
If this "theorem" is not true, I would appreciate any pointer to a weaker result that is true.
Finally, I expect that the word "descent" will be used by any human answering this question, and I want to indicate that, while I am a novice, I'm not entirely ignorant of this general theory. I've spent the last week on a safari trying to find a basic reference for descent theory in the context of complex analytic spaces, and currently I still have mostly empty hands.
In the complement of my empty hands, there are some previous MO questions that touch on the issue of complex analytic descent, but I haven't found a plug and play result that answers this question. Therefore, I would appreciate any comment, idea, or reference that clarifies this issue.
