Holomorphic Fiber Bundles over Riemann Surfaces

For the purpose of this paper a fiber bundle F—>X over a Riemann surface X is meant to be a fiber bundle in the sense of N. Steenrod [62] where the base space is X, the fiber a complex space, the structure group G a complex Lie group that acts as a complex transformation group on the fiber, and the transition functions g%j{x) are holomorphic mappings into G. Correspondingly, cross-sections are assumed to be holomorphic cross-sections. We shall use freely the notations of [62], Whenever we report about families of fiber bundles we mean holomorphic families of fiber bundles; the basic notations concerning families of fiber bundles can be found in [30 ] and shall also be used freely. Triviality of bundles resp. families of bundles is always supposed to be holomorphic triviality.