An equivariant version of Grauert's Oka principle

Abstract

For example, a theorem of Oka asserts that every holomorphic line bundle over a domain of holomorphy in (12" is holomorphicaIly trivial if and only if it is topologically trivial. In [G3] Grauer t proves that the category of bundles over a Stein space X with a complex Lie group as a structure group is the same up to isomorphism if considered in the topological or complex analytic category. We would like to underl ine that Grauert ' s method of proof also has far reaching consequences (see e.g. [FR1], [FR2]). In this article we prove the equivariant version of Grauert ' s Oka Principle for a compact Lie group of holomorphic t ransformations on X. The precise statements can be found in w w and w For example, as a consequence of the main results we obtain: