Q-pseudoconvex and q-holomorphically convex domains

TL;DR

This paper establishes new links between q-pseudoconvexity and q-holomorphic convexity in complex Euclidean spaces, proving that certain pseudoconvex domains are also holomorphically convex under specific conditions.

Contribution

It generalizes Basener's theorem by showing that smoothly bounded strictly q-pseudoconvex domains are (q+1)-holomorphically convex and identifies conditions under which they are q-holomorphically convex.

Findings

Strictly q-pseudoconvex domains are (q+1)-holomorphically convex.

Additional assumptions lead to q-holomorphic convexity.

Any open subset of complex n-space is n-holomorphically convex.

Abstract

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q-pseudoconvexity and q-holomorphic convexity: we prove that any smoothly bounded strictly q-pseudoconvex open subset of the complex Euclidean space is (q + 1)-holomorphically convex; moreover, assuming that the given open set verifies an additional assumption, we prove that it is q-holomorphically convex. We also prove that any open subset of the complex n-dimensional Euclidean space is n-holomorphically convex.