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 non-elliptic 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 $\mathbb{C}^n\backslash \mathbb{R}^k$ of a totally real affine subspace is Oka if $n > 1$ and $(n,k) \ne (2,1),(2,2),(3,3)$.