Equivariant complex extensions of homogeneous spaces
Let \(M\) be a real analytic manifold of dimension \(n\). By its complex extension we mean a pair \((\Omega, \varphi)\), where \(\Omega\) is a complex manifold of complex dimension \(n\) and \(\varphi\) is a real analytic isomorphism of the manifold \(M\) onto a closed totally real submanifold \(\varphi (M) \subset \Omega\). It is known that if \(M\) is paracompact, then a complex extension always exists. Moreover, as \textit{H. Grauert} has proved [Ann. Math., II. Ser. 68, No. 2, 460-472 (1958; Zbl 0108.078)], we can take a Stein manifold for \(\Omega\). Assume that the analytic action of a real Lie group \(G\) is given on \(M\). We say that a complex extension \((\Omega, \varphi)\) is \(G\)-equivariant if the analytic action of the group \(G\) on \(\Omega\) is defined, where each transformation \(g : \Omega \to \Omega\) \((g \in G)\) is holomorphic and the mapping \(M @>> \sim > \varphi (M) \subset \Omega\) commutes with the action of \(G\). The goal of the present article is twofold. First, we prove that equivariant extensions do not always exist, even if \(M\) is a homogeneous space of the group \(G\). Second, for homogeneous spaces of linear groups with a compact stationary subgroup that have obvious equivalent complex extensions, we construct equivariant Stein extensions with proper actions.
scientific article; zbMATH DE number 525686