The following is motivated by a (now-deleted) MSE-question by @aglearner.


Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the group $G= {\mathbb C}^\times$, and suppose that $G\times X\to X$ is an algebraic action. Assume, in addition, that each $G$-orbit in $X$ is closed.


**Question 1.** Is the quotient $X/G$ (in the sense of general topology) Hausdorff?


(Note that I am *not* taking the GIT quotient here.)


Hausdorffness of the quotient fails if I relax the assumption on $X$ to that of a *quasi-affine* subvariety in the following standard example: Take $X={\mathbb C}^2\setminus \{(0,0)\}$ and consider the action given by
$$
(t, (x,y)) \mapsto (tx, t^{-1}y), \quad t\in G, (x,y)\in X. 
$$


The next question is a complex-analytic version of Question 1:


**Question 2.** Suppose that $X\subset {\mathbb C}^n$ is a Stein submanifold, $G$ is as above and $G\times X\to X$ is a holomorphic action with closed orbits. Is $X/G$ Hausdorff?


I expect answers to both questions to be negative (and counter-examples given by a smooth affine subvariety $X$) but cannot think of any examples.



Group actions on affine varieties with closed orbits
I think the answer to Question 1 is yes, since under your hypothesis the set-theoretical quotient *is* essentially the GIT quotient. This follows from the fact that $G$-invariant regular functions on $X$ separate orbits (when these are closed).


There is an analytic proof of this fact. Let $V$ and $W$ be two disjoint closed orbits. Then 1 can be written as the sum of two regular functions $f\_V$ and $f\_W$ vanishing respectively on $V$ and $W$ (because $I(V) + I(W) = \mathbb C[X]$). In particular, $f\_V = 0$ on $V$ and $f\_V = 1$ on $W$. After averaging $f\_V$ under the action of the unit circle $U\_1\subset \mathbb C^\times$, one can assume that $f\_V$ is $U\_1$-invariant. Since $f\_V$ is holomorphic, the action of $\mathbb C^\times$ is holomorphic and $U\_1$ is a real form of $\mathbb C^\times$, this implies that $f\_V$ is in fact $\mathbb C^\times$-invariant.


My impression is that one could do the same with holomorphic functions on Stein manifolds. My only concern is about the existence of a holomorphic function which is $0$ on $V$ and $1$ on $W$. (This condition ensures that, in the averaging process, $f\_V$ remains non-zero on $W$.)


