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Zbl 0794.32022
Heinzner, Peter
Equivariant holomorphic extensions of real analytic manifolds. (English)
[J] Bull. Soc. Math. Fr. 121, No.3, 445-463 (1993). ISSN 0037-9484

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 $X\sp*$ equipped with a holomorphic action of the complexified Lie group $G\sp \bbfC$ of $G$ and a real analytic $G$-map $i:X \to X\sp*$ with the following universal property: for every complex space $Z$ where $G\sp \bbfC$ acts holomorphically and every real analytic $G$-map $\varphi:X \to Z$ there exists a holomorphic $G\sp \bbfC$-map $\varphi\sp*$ defined on a $G\sp \bbfC$-invariant neighborhood of $i(X)$ in $X\sp*$ such that $\varphi=\varphi\sp*i$. Moreover if $G$ is a holomorphically extendable Lie group, then $X\sp*$ is smooth and $i$ is a closed embedding.\par It is also proved that the quotient space $Q\sp*$ of $X\sp*$ 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 $X/G$.
[A.Tancredi (Perugia)]

MSC 2000:: *32M05 Automorphism groups of complex spaces
32C05 Real-analytic manifolds and spaces
32V40 Real submanifolds in complex manifolds

Keywords: Lie group; analytic action; complexification

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