Oka's principle for holomorphic fiber bundles with sprays

Details Oka's principle for holomorphic fiber bundles with sprays Oka's principle for holomorphic fiber bundles with sprays

2003-05-16

arxiv.org/abs/math/0305238v1

Math. Ann., 317 (2000), 117-154

Mathematics

Complex Variables

Scientific paper

We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction see the papers math.CV/0101040, math.CV/0101040, math.CV/0101034, math.CV/0101238, math.CV/0107039, math.CV/0110201.

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