Let (E, h) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with \(\dim X>1\) . Assume that for each point \(x\in X\) there exists a divisor \(D\in |E|\) in the complete linear system determined by E whose complement \(X\setminus D\) is a Stein neighbourhood of x with the density property. Then, the disc bundle \(\Delta _h(E)=\{e\in E:|e|_h<1\}\) is an Oka manifold while \(D_h(E)=\{e\in E:|e|_h>1\}\) is a Kobayashi hyperbolic domain. In particular, the zero section of E admits a basis of Oka neighbourhoods \(\{|e|_h<c\}\) with \(c>0\) . We show that this holds if X is a rational homogeneous manifold of dimension \(>1\) . This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.
