Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds

TL;DR

This paper constructs holomorphic families of Fatou-Bieberbach domains to demonstrate that complements of polynomially convex sets in complex Euclidean spaces are Oka manifolds, extending to Stein manifolds with the density property.

Contribution

It provides a new, simplified proof that complements of polynomially convex sets are Oka manifolds, and extends the result to Stein manifolds with the density property.

Findings

Complements of polynomially convex sets in ^n are Oka manifolds.

Construction of holomorphic families of Fatou-Bieberbach domains.

Extension of results to Stein manifolds with the density property.

Abstract

We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of Yuta Kusakabe to the effect that the complement $\mathbb C^n\setminus K$ of any polynomially convex subset $K$ of $\mathbb C^n$ is an Oka manifold. The analogous result is obtained with $\mathbb C^n$ replaced by any Stein manifold with the density property.