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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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