Exercise 26, § 6

Let G be an n-dimensional connected compact complex Lie group, X a finite-dimensional connected complex analytic manifold and $(g, x) \mapsto gx$ a law of analytic left operation of G on X. For all $g \in G$, let $\varphi(g)$ be the mapping $x \mapsto gx$ of X into X. Suppose, that, for all $g \in G$ distinct from e, $\varphi(g) \neq \mathrm{Id}_X$. Then every orbit of G in X is an n-dimensional closed submanifold of X. (Let $x \in X$, H be the stabilizer of x and H' the identity component of H. For all $g \in H'$, let $u(g)$ be the tangent mapping at x to $\varphi(g)$. Arguing as in Proposition 6 of no. 3, show that $u(g)$ is the identity for all $g \in H'$. Using § 1, Exercise 4 and the connectedness of X, deduce that $H' = \{e\}$.