Group actions on affine varieties with closed orbits

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.

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 $$W$$.)

• 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) Nov 9 '21 at 14:11
• 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. Nov 9 '21 at 14:57
• Thank you Nicolas and Richard! Nov 16 '21 at 22:18

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).