Flexibility Properties of Complex Manifolds and Holomorphic Maps

Abstract

This chapter is new in the current edition and focuses on recent developments. We begin by comparing the Oka property with other standard holomorphic flexibility properties of complex manifolds that have been studied in the literature. We introduce the class of stratified Oka manifolds and show that every such manifold is strongly dominable by the complex Euclidean space of the same dimension. Next we describe what we know about which compact complex surfaces are Oka. We also introduce and study the class of Oka maps. In a section contributed by Finnur Larusson we give a homotopy-theoretic viewpoint of modern Oka theory. We conclude by discussing miscellaneous recent results and collecting open problems.