Oka domains in Euclidean spaces

Abstract

In this paper we find surprisingly small Oka domains in Euclidean spaces ${\mathbb{C}}^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in ${\mathbb{C}}^n$ we show that ${\mathbb{C}}^n\setminus E$ is an Oka domain. In particular, there are Oka domains which are only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives families of smooth real hypersurfaces ${\Sigma }_t\subset {\mathbb{C}}^n$ $(t\in \mathbb{R})$ dividing ${\mathbb{C}}^n$ in an unbounded hyperbolic domain and an Oka domain such that at the threshold value $t=0$ the hypersurface ${\Sigma }_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in ${\mathbb{C}}^n$ for $n>1$ whose projective closure $\overline{E}\subset \mathbb{C}{\mathbb{P}}^n$ avoids a hyperplane $\Lambda \subset \mathbb{C}{\mathbb{P}}^n$ and is polynomially convex in $\mathbb{C}{\mathbb{P}}^n\setminus \Lambda \cong {\mathbb{C}}^n$, then ${\mathbb{C}}^n\setminus E$ is an Oka domain.

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Cited by in corpus (4)

• Recent developments on Oka manifolds
• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
• Oka tubes in holomorphic line bundles
• Proper holomorphic embeddings with small limit sets