Equivariant holomorphic extensions of real analytic manifolds. (English) Zbl 0794.32022

Heinzner, Peter

Bull. Soc. Math. Fr. 121, No. 3, 445-463 (1993).

The author is interested in the equivariant complexification of a real analytic manifold \(X\) equipped with a proper analytic action of a Lie group \(G\). He proves that there exist a complex space \(X^*\) equipped with a holomorphic action of the complexified Lie group \(G^ \mathbb{C}\) of \(G\) and a real analytic \(G\)-map \(i:X \to X^*\) with the following universal property: for every complex space \(Z\) where \(G^ \mathbb{C}\) acts holomorphically and every real analytic \(G\)-map \(\varphi:X \to Z\) there exists a holomorphic \(G^ \mathbb{C}\)-map \(\varphi^*\) defined on a \(G^ \mathbb{C}\)-invariant neighborhood of \(i(X)\) in \(X^*\) such that \(\varphi=\varphi^*i\). Moreover if \(G\) is a holomorphically extendable Lie group, then \(X^*\) is smooth and \(i\) is a closed embedding.
It is also proved that the quotient space \(Q^*\) of \(X^*\) with respect to the smallest complex analytic equivalent relation given by the \(G\)- orbits is a Stein space that can be considered as a natural complexification of the semianalytic space \(X/G\).

Reviewer: A.Tancredi (Perugia)

MSC:

32M05	Complex Lie groups, group actions on complex spaces
32C05	Real-analytic manifolds, real-analytic spaces
32V40	Real submanifolds in complex manifolds

Keywords:

Lie group; analytic action; complexification

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