Orbit dimension and closed orbits for complex group actions

For any complex affine algebraic group $K$, [F5] and [F7] imply that distinct irreducible components are disjoint. The finitely many components are therefore open and closed; since each is irreducible and hence connected, they are exactly the connected components. Let $C$ be the component containing the identity. Translation by $c\in C$ carries the unique component through the identity onto the unique component through $c$, so $cC=C$. Inversion and conjugation fix the identity and permute components, hence preserve $C$. Thus $C=K^{\circ }$ is a closed normal irreducible open subgroup, and translation by any $k\in K$ identifies $C$ with the component through $k$. Its finitely many cosets are precisely the components, all of the same dimension; [F4] gives $\dim K=\dim K^{\circ }$. Apply this also to the closed classical subgroup $H=G_x$: its identity component $H^{\circ }$, being connected and containing the identity, lies in $G^{\circ }$. Since $H\cap G^{\circ }$ is open and closed in $H$, it is a nonempty union of components of $H$, each of dimension $\dim H$. Hence [F4] gives $\dim (G^{\circ }{)}_x=\dim (H\cap G^{\circ })=\dim H=\dim G_x$.

Suppose first that $G$ is connected. Then $G_x=(G^{\circ }{)}_x$ is a closed subgroup by [F1], and [F2], read through [F5], makes $Gx$ a locally closed subvariety. By step 1.1 the connected group $G$ is irreducible, so its image under the surjective orbit map is irreducible. The connected dimension supplier in [F2] gives $\dim Gx=\dim G-\dim G_x$, hence $\dim G=\dim G_x+\dim Gx$.

Still with $G$ connected, put $Z=\overline{Gx}$. By step 2.1, $Gx$ is irreducible and locally closed, hence open dense in $Z$. Thus $Z$ is irreducible and $\dim Z=\dim Gx$ by [F6]. The orbit-closure clause of [F2] makes every orbit in $Z\setminus Gx$ strictly smaller in dimension. This proves the connected case of (b).

For arbitrary $G$, step 1.1 writes $Gx$ as finitely many distinct pairwise disjoint $G^{\circ }$-orbits $O_i=g_i{G}^{\circ }x$, permuted transitively by $G$. That step also gives $\dim (G^{\circ }{)}_x=\dim G_x$ and $\dim G=\dim G^{\circ }$. Hence every $O_i$ has dimension $d=\dim G^{\circ }-\dim G_x=\dim G-\dim G_x$ by step 2.1 and translation. Each $O_i$ is irreducible and locally closed, so it is open dense in its irreducible closure $Z_i$, with $\dim Z_i=d$ by step 2.1. If $i\mathrm{\ne }j$ and $O_i\cap Z_j\mathrm{\ne }\mathrm{\varnothing }$, the $G^{\circ }$-stability of $Z_j$ implies $O_i\subseteq Z_j$ and then $Z_i\subseteq Z_j$. Equal dimensions and [F6] force $Z_i=Z_j$, whose two nonempty open subsets $O_i,O_j$ would intersect, a contradiction. Thus $O_i\cap Z_j=\mathrm{\varnothing }$ for $i\mathrm{\ne }j$. Consequently $Z=\overline{Gx}={\bigcup }_i{Z}_i$ has $Z\setminus Gx={\bigcup }_i(Z_i\setminus O_i)$ closed, and $O_i=Z_i\cap Gx$ is closed in $Gx$. By [F4], $\dim Gx=d$; the irreducible components of $Z$ are exactly the distinct $Z_i$, each of dimension $d$. Every $G^{\circ }$-orbit in the boundary has dimension less than $d$ by [F2]. For any full $G$-orbit there, apply the same finite-union construction to its finitely many connected-group orbits, which are translates of one another: its dimension is their common dimension, also less than $d$. This proves (a) and (b).

If $Gx$ has minimal dimension among the orbits in $X$, step 4.1 leaves no boundary orbit of smaller dimension, so $Gx$ is closed. For an arbitrary orbit closure $\overline{Gz}$, choose an orbit $Gy$ in it with least dimension, which exists because the nonempty set of orbit dimensions is a subset of the nonnegative integers. This closure is closed and $G$-stable, so $\overline{Gy}\subseteq \overline{Gz}$. A boundary orbit of $Gy$ would have smaller dimension by step 4.1 and still lie in $\overline{Gz}$, contradicting the choice. Thus $Gy$ is closed, proving (c).

Steps 4.1 and 5.1 prove all the stated conclusions, with the inherited Axiom of Choice. The component argument was proved locally in step 1.1 using the stated regular-point and finite-component suppliers.