Oka tubes in holomorphic line bundles

Let ( E , h ) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with X>1 dim X > 1 . Assume that for each point x∈ X x ∈ X there exists a divisor D∈ |E| D ∈ | E | in the complete linear system determined by E whose complement X D X \ D is a Stein neighbourhood of x with the density property. Then, the disc bundle Δ ₕ(E)=∈ E:|e|ₕ<1\ Δ h ( E ) = { e ∈ E : | e | h < 1 } is an Oka manifold while Dₕ(E)=∈ E:|e|ₕ>1\ D h ( E ) = { e ∈ 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\ { | e | h < c } with $$c>0$$ c > 0 . We show that this holds if X is a rational homogeneous manifold of dimension $$>1$$ > 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.