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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.

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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 $W$.)

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  • $\begingroup$ If the argument in the Stein manifold case only relies on the existence of a function which is 0 on W and 1 on V, then this is indeed possible, cf., for example mathoverflow.net/q/383638 (which is stated for X=C^n, but the same argument applies for any Stein manifold X) $\endgroup$ Nov 9 '21 at 14:11
  • $\begingroup$ Great! Then I think we're good. After averaging, you can make this function $\mathbb U_1$-invariant, hence $\mathbb C^\times$-invariant. So invariant holomorphic functions separate points, implying that the largest Hausdorff quotient is the set theoretical quotient. $\endgroup$ Nov 9 '21 at 14:57
  • $\begingroup$ Thank you Nicolas and Richard! $\endgroup$ Nov 16 '21 at 22:18
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Let $X$ be a complex affine algebraic set with the analytic topology and let $G$ be a complex affine algebraic group acting rationally on $X$.

$\newcommand\sslash{/\hspace{-0.2ex}/}$Let $X\sslash G=\operatorname{Spec}_\text{max}\mathbb{C}[X]^G$ be the affine GIT quotient of $X$ by $G$. Give $X\sslash G$ the analytic topology. So $X\sslash G$ is necessarily Hausdorff.

Let $X^*\subset X$ be the subspace of points with closed $G$-orbits (so $X^*=X$ if all orbits are closed). Give $X^*/G$ the quotient topology, as with $X/G$.

Then $X\sslash G$ is homeomorphic to $X^*/G$ (this follows from work of Luna), and homotopic to $X/G$ (Proposition 3.4 of Florentino, Lawton, and Ramras - Homotopy Groups of Free Group Character Varieties).

So, in particular, the answer to your first question is yes. I am not sure about your second question.

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    $\begingroup$ Thank you, Sean! I wish I could accept both proofs, but I will accept Nicolas' answer since, as explained by Richard, it can be adopted to the general Stein setting. $\endgroup$ Nov 16 '21 at 22:17
  • $\begingroup$ I completely understand :) If you need to cite something though in the affine case, I have an elementary proof here (Theorem 2.1): arxiv.org/pdf/1301.7616.pdf. $\endgroup$ Nov 16 '21 at 23:53

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