Oka tubes in holomorphic line bundles

TL;DR

The paper demonstrates that certain disc bundles in semipositive holomorphic line bundles over complex manifolds are Oka manifolds, while their complements are Kobayashi hyperbolic, linking Oka properties to metric positivity.

Contribution

It establishes conditions under which disc bundles are Oka manifolds and their complements are hyperbolic, especially for rational homogeneous manifolds, connecting Oka properties to metric positivity.

Findings

Disc bundles are Oka manifolds under certain conditions.

Complements of these bundles are Kobayashi hyperbolic.

Results apply to rational homogeneous manifolds like projective spaces.

Abstract

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.