Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910791031062528 |
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| author | Fornæss, John Erik Pal, Ratna |
| author_facet | Fornæss, John Erik Pal, Ratna |
| contents | For $d\geq 2$, we discuss $d$-dimensional complex manifolds $M$ that are the increasing union of bounded open sets $M_n$'s of $\mathbb{C}^d$ with a common uniform squeezing constant. The description of $M$ is given in terms of the corank of the infinitesimal Kobayashi metric of $M$, which is shown to be identically constant on $M$. The main result of this article says that if $M$ has full Kobayashi corank, then $M$ can be written as an increasing union of the unit ball; if $M$ has zero Kobayashi corank, then $M$ has a bounded realization with a uniform squeezing constant; and if $M$ has an intermediate Kobayashi corank, then $M$ has a local weak vector bundle structure. The above description of $M$ is used to show that the dimension of the Bergman space of $M \subseteq \mathbb{C}^d$ is either zero or infinity. This settles Wiegerinck's conjecture for those pseudoconvex domains in higher dimensions that are increasing union of bounded domains with a common uniform squeezing constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces Fornæss, John Erik Pal, Ratna Complex Variables Primary: 32F45, Secondary: 32H02, 32A36 For $d\geq 2$, we discuss $d$-dimensional complex manifolds $M$ that are the increasing union of bounded open sets $M_n$'s of $\mathbb{C}^d$ with a common uniform squeezing constant. The description of $M$ is given in terms of the corank of the infinitesimal Kobayashi metric of $M$, which is shown to be identically constant on $M$. The main result of this article says that if $M$ has full Kobayashi corank, then $M$ can be written as an increasing union of the unit ball; if $M$ has zero Kobayashi corank, then $M$ has a bounded realization with a uniform squeezing constant; and if $M$ has an intermediate Kobayashi corank, then $M$ has a local weak vector bundle structure. The above description of $M$ is used to show that the dimension of the Bergman space of $M \subseteq \mathbb{C}^d$ is either zero or infinity. This settles Wiegerinck's conjecture for those pseudoconvex domains in higher dimensions that are increasing union of bounded domains with a common uniform squeezing constant. |
| title | Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces |
| topic | Complex Variables Primary: 32F45, Secondary: 32H02, 32A36 |
| url | https://arxiv.org/abs/2407.02130 |