Strong Stein neighborhood bases

TL;DR

This paper characterizes when a smooth bounded pseudoconvex domain in complex space has a strong Stein neighborhood basis, showing invariance under certain holomorphic maps, which aids understanding of complex geometric structures.

Contribution

It provides multiple characterizations for strong Stein neighborhood bases and proves their invariance under proper holomorphic maps extending smoothly to the boundary.

Findings

Several characterizations for strong Stein neighborhood bases.

Invariance of the condition under proper holomorphic maps.

Extension of the concept to boundary-smooth maps.

Abstract

Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\in C^n:r(z)<a} is pseudoconvex for sufficiently small a>0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary.