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Oka tubes in holomorphic line bundles

Franc Forstneric, Yuta Kusakabe

TL;DR

The paper shows that for a polarised pair (X,E) with a semipositive Hermitian line bundle (E,h) over a compact X of dimension > 1, the disc bundle Δ_h(E) is an Oka manifold while its exterior tube D_h(E) is Kobayashi hyperbolic, provided that every x ∈ X lies in a divisor D ∈ |E| with X \ D Stein and possessing the density property. This links Oka flexibility to metric positivity and extends the class of known Oka manifolds by exploiting Hartogs-domain techniques and Kusakabe’s density-property localization; the key new ingredient is the polarised density property, which holds for all polarised manifolds built from CP^n, Grassmannians, and, more generally, rational homogeneous manifolds of dimension > 1. The results include detailed consequences for positive and negative line bundles, tensor powers, duals, and products, showing that many natural polarised geometries admit Oka tubes and hyperbolic exterior domains. In addition, the paper derives holomorphic mapping consequences: from Stein S into E with dim S < dim E one can construct proper maps into D_h(E) (under Griffiths negativity), or maps whose cluster set lies in the zero section when Δ_h(E) is Oka, thereby illustrating a concrete interaction between Oka theory and positivity-driven rigidity. Overall, the work provides a metric-positivity lens on the Oka principle and expands the toolkit for generating flexible complex-analytic structures from positively curved geometric data.

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 $Δ_h(E)=\{e\in E:|e|_h<1\}$ 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.

Oka tubes in holomorphic line bundles

TL;DR

The paper shows that for a polarised pair (X,E) with a semipositive Hermitian line bundle (E,h) over a compact X of dimension > 1, the disc bundle Δ_h(E) is an Oka manifold while its exterior tube D_h(E) is Kobayashi hyperbolic, provided that every x ∈ X lies in a divisor D ∈ |E| with X \ D Stein and possessing the density property. This links Oka flexibility to metric positivity and extends the class of known Oka manifolds by exploiting Hartogs-domain techniques and Kusakabe’s density-property localization; the key new ingredient is the polarised density property, which holds for all polarised manifolds built from CP^n, Grassmannians, and, more generally, rational homogeneous manifolds of dimension > 1. The results include detailed consequences for positive and negative line bundles, tensor powers, duals, and products, showing that many natural polarised geometries admit Oka tubes and hyperbolic exterior domains. In addition, the paper derives holomorphic mapping consequences: from Stein S into E with dim S < dim E one can construct proper maps into D_h(E) (under Griffiths negativity), or maps whose cluster set lies in the zero section when Δ_h(E) is Oka, thereby illustrating a concrete interaction between Oka theory and positivity-driven rigidity. Overall, the work provides a metric-positivity lens on the Oka principle and expands the toolkit for generating flexible complex-analytic structures from positively curved geometric data.

Abstract

Let be a semipositive hermitian holomorphic line bundle on a compact complex manifold with . Assume that for each point there exists a divisor in the complete linear system determined by whose complement is a Stein neighbourhood of with the density property. Then, the disc bundle is an Oka manifold while is a Kobayashi hyperbolic domain. In particular, the zero section of admits a basis of Oka neighbourhoods with . We show that this holds if is a rational homogeneous manifold of dimension . 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.
Paper Structure (5 sections, 25 theorems, 37 equations)

This paper contains 5 sections, 25 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main results
  4. Examples of line bundles satisfying Theorem \ref{['th:density2']}
  5. Holomorphic maps from Stein manifolds to vector bundles

Key Result

Theorem 1.1

Given a positive holomorphic line bundle $E=\mathscr{O}_{\mathbb{CP}^n}(k)$ on $\mathbb{CP}^n$$(n\ge 1,\ k\ge 1)$ and a semipositive hermitian metric $h$ on $E$ (i.e., $\mathfrak{i} \Theta_h\ge 0$), the following assertions hold. For a negative holomorphic line bundle $E=\mathscr{O}_{\mathbb{CP}^n}(k)$$(n\ge 1,\ k\le -1)$ and a seminegative hermitian metric $h$ on $E$ ($\mathfrak{i} \Theta_h\le 0

Theorems & Definitions (54)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 44 more