Oka tubes in holomorphic line bundles

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.

This version agrees with the open access published version in Math. Ann

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• Recent developments on Oka manifolds
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Cited by in corpus (1)

• Oka-1 manifolds: New examples and properties