Stable points of an affine action

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a complex affine algebraic group acting algebraically on an affine algebraic set $X$ (Classical complex affine algebraic actions and rational modules). A point $x\in X$ is stable if

(i) its orbit $Gx$ is closed in $X$, and

(ii) its stabilizer $G_x$ is finite, equivalently $\dim G_x=0$ (Orbit dimension and closed orbits for complex group actions, Global and local dimension of classical varieties);

Here $G_x$ is a closed subgroup of the finite-type group $G$ and has finitely many irreducible components (Classical varieties have finite irreducible decompositions). If its dimension is zero, each component is a single point: otherwise a closed point strictly contained in that component would give a chain of length one. Conversely a finite set of closed points has dimension zero. Thus $G_x$ is finite exactly when its dimension is zero. The stable locus $X^s\subseteq X$ is the set of stable points, and the unstable locus is its complement.

Stability implies that the orbit is closed; the converse fails: the trivial action of ${\mathbf{G}}_m$ on a point has closed orbit but positive-dimensional stabilizer. Stability is preserved by replacing $G$ with $G^{\circ }$, because $G_x$ and $(G^{\circ }{)}_x$ have the same dimension by Orbit dimension and closed orbits for complex group actions, while the $G$-orbit of a point is a finite union of $G^{\circ }$-orbits permuted by $G$. Those $G^{\circ }$-orbits are closed in $Gx$ by Proof 3.1 of the orbit lemma, so a closed $G$-orbit makes each of them closed in $X$. Conversely, if $G^{\circ }x$ is closed in $X$, its finitely many translates have closed union $Gx$.

• This is Brion's Definition 1.25 (printed p. 9) with the finite-stabilizer condition expressed by dimension, using that a closed subgroup of a finite-type complex algebraic group is finite if and only if it is zero-dimensional. The final strictness remark is Brion Example 1.27(1) and is made explicit as a counterexample on the companion page.
• The definition is choice-free; the dimensional restatement and the $G^{\circ }$-reduction inherit AC from the orbit lemma above. No closedness of $X$ or of the stabilizer is assumed beyond what the named suppliers give.