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 polynomially convex set in Cn(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
CnβRk of a totally real affine subspace is
Oka if n>1 and (n,k)ξ =(2,1),(2,2),(3,3).Comment: 15 page