Oka properties of complements of holomorphically convex sets

Abstract

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in ${\mathbb{C}}^n$ $(n>1)$ is Oka. Furthermore, we obtain new examples of nonelliptic Oka manifolds which negatively answer Gromov's question. The relative version of the main theorem is also proved. As an application, we show that the complement ${\mathbb{C}}^n\setminus {\mathbb{R}}^k$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\ne (2,1),(2,2),(3,3)$.

arXiv commentAnn. of Math. (2), to appear

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• On nonlinear Rudin-Carleson type theorem