Let $(E,h)$ be a semipositive hermitian holomorphic line bundle on a compact complex manifold $X$. 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 $E\setminus \overline{\Delta_h(E)}=\{e\in E:|e|_h>1\}$ is a pseudoconvex Kobayashi hyperbolic domain. In particular, the zero section of $E$ admits a basis of Oka neighbourhoods $\{|e|_h<c\}$ with $c>0$. This holds for any nontrivial semipositive hermitian holomorphic line bundle on a projective space or a Grassmannian of dimension $>1$, as well as on many other compact projective manifolds.
This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds. (Joint work with Yuta Kusakabe, Kyoto University.)