## An equivariant version of Grauert’s Oka principle.(English)Zbl 0837.32004

An equivariant version of Grauert’s Oka principle for a compact Lie group of holomorphic transformations of a Stein space is proved in this paper. Let $$K$$ be a compact Lie group and $$X$$ be a Stein $$K$$-space. Fix a Kempf- Ness subset $$R$$ of $$X$$ with respect to a $$K$$-invariant strictly plurisubharmonic exhaustion function. Let $$X//K$$ be the categorial quotient. The major object in this version is $$Q(R)$$ which is defined to be the sheaf of groups over $$X//K$$.
The main results are the following three theorems.
Connectedness Theorem: The group $$Q(R) (X//K)$$ is arcwise connected;
Runge Theorem: If $$U$$ is Runge in $$X//K$$, then the image of $$Q(R) (X//K)$$ in $$Q(R) (U)$$ is dense;
Vanishing Theorem: The first cohomology group $$H^1 (X//K, {\mathcal Q} (R)) = 0$$.
There are also corresponding theorems for analytic cubes of dimension $$d$$. These theorems have many consequences. For example, it is proved here that if $$K$$ is a compact Lie group of holomorphic transformations of a reduced Stein space $$X$$, then every topological complex $$K$$-vector bundle over $$X$$ is $$K$$-equivariantly isomorphic to a holomorphic one, and two holomorphic ones are $$K$$-equivariantly holomorphically isomorphic if and only if they are $$K$$-equivariantly topologically isomorphic.

### MSC:

 32C15 Complex spaces 32E10 Stein spaces
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### References:

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