Applications of Oka Theory and Its Methods

Abstract

In this chapter we apply the methods and results of Oka theory to a variety of problems in Stein geometry. In particular, we discuss the structure of holomorphic vector bundles and their homomorphisms over Stein spaces, find the minimal number of generators of coherent analytic sheaves over Stein spaces, consider the problem of complete intersections and, more generally, of elimination of intersections of holomorphic maps with complex subvarieties, present the solution of the holomorphic Vaserstein problem, discuss transversality theorems for holomorphic and algebraic maps from Stein manifolds, and obtain estimates on the dimension of singularity (degeneration) sets of generic holomorphic maps from Stein manifolds to Oka manifolds. In the last part of the chapter we further develop the method of local holomorphic sprays of maps from strongly pseudoconvex Stein domains. We apply this technique to prove an up-to-the-boundary version of the Oka principle on such domains, and we establish a Banach manifold structure theorem for certain spaces of holomorphic maps from strongly pseudoconvex domains to arbitrary complex manifolds.