Oka Manifolds

Abstract

The heuristic Oka principle states that analytic problems on Stein spaces which can be cohomologically formulated have only topological obstructions. The main example is the classical theorem of Oka and Grauert stating that the topological classification of principal fiber bundles over Stein spaces agrees with their holomorphic classification. The chapter begins with an introduction to the Oka-Grauert principle and its reduction to the problem of deforming continuous sections to holomorphic sections in principal fiber bundles. This naturally leads to the theory of Oka manifolds, a class of complex manifolds that has recently emerged from the modern theory of the Oka principle. The main result is that sections of any stratified holomorphic fiber bundle with Oka fibers over a reduced Stein space satisfy all forms of the Oka principle.