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

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 $$X/G$$.

### 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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### References:

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