Oka tubes in holomorphic line bundles
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915234122301440 |
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| author | Forstneric, Franc Kusakabe, Yuta |
| author_facet | Forstneric, Franc Kusakabe, Yuta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14871 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Oka tubes in holomorphic line bundles Forstneric, Franc Kusakabe, Yuta Complex Variables Primary 32Q56. Secondary 32E10, 32L05, 32Q10 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. |
| title | Oka tubes in holomorphic line bundles |
| topic | Complex Variables Primary 32Q56. Secondary 32E10, 32L05, 32Q10 |
| url | https://arxiv.org/abs/2310.14871 |