Oka properties of ball complements

TL;DR

This paper establishes the Oka property for holomorphic maps into complements of convex sets in complex Euclidean spaces, extending classical results and providing new constructions of proper holomorphic maps with controlled range.

Contribution

It proves the Oka property for maps into ball complements in complex spaces and constructs proper holomorphic maps with additional range control, extending classical results.

Findings

Holomorphic maps from Stein manifolds into complements of convex sets satisfy the Oka property.

The Oka property holds for polynomially convex complements when 2*dim X < n.

Constructs proper holomorphic maps, immersions, and embeddings with controlled range.

Abstract

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X < n$. We also construct proper holomorphic maps, immersions and embeddings $X\to\mathbb{C}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.