  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 $(n,k)\neq(2,1),(2,2),(3,3)$.
