Stein Neighborhoods and Holomorphic Approximation

Abstract

The major part of this chapter is devoted to the construction of open Stein neighborhoods of sets in arbitrary complex spaces. Highlights include Siu’s theorem on Stein neighborhoods of Stein subvarieties and some generalizations, the Docquier-Grauert type theorems on holomorphic retractions onto Stein submanifolds, and the construction of Stein neighborhoods of compact holomorphically convex sets with attached totally real handles. We also prove certain extension and approximation theorems for holomorphic mappings, analyze the geometry of Morse critical points of strongly plurisubharmonic and q-convex functions, and consider the topological structure of Stein spaces.