Stein neighborhood bases of embedded strongly pseudoconvex domains and approximation of mappings

TL;DR

This paper extends Siu's theorem by constructing Stein neighborhood bases for strongly pseudoconvex subvarieties in complex spaces, including higher dimensions and complex curves, and applies these results to holomorphic mapping approximation.

Contribution

It generalizes existing results to higher dimensions and complex subvarieties, providing new Stein neighborhood bases and approximation techniques.

Findings

Constructed Stein neighborhood bases for strongly pseudoconvex subvarieties.

Extended Siu's theorem to higher-dimensional cases.

Applied results to approximation of holomorphic mappings.

Abstract

In this paper we construct a Stein neighborhood basis for any compact subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior $A\backslash bA$ in a complex space $X$. This is an extension of a well known theorem of Siu. When $A$ is a complex curve, our result coincides with the result proved by Drinovec-Drnov\v{s}ek and Forstneri\v{c}. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly's proof of Siu's theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results are applied to the approximation problem for holomorphic mappings.