$L^2$ extension of holomorphic functions and log canonical places
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909598692147200 |
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| author | Kim, Dano Wang, Xu |
| author_facet | Kim, Dano Wang, Xu |
| contents | In an influential $L^2$ extension theorem due to Demailly, the finiteness of an $L^2$ norm called the Ohsawa norm determines whether a given holomorphic function can be extended. This result has been further generalized by Zhou and Zhu to the case when the quasi-plurisubharmonic defining function of the subvariety has non-analytic singularities. We show that, however, there exist many instances of such defining functions for which only the zero function has finite Ohsawa norm, so that the $L^2$ extension statement is void in such cases, even when it has a unique log canonical place. Such a defining function occurs already among some of the simplest non-analytic singularities, namely toric ones. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^2$ extension of holomorphic functions and log canonical places Kim, Dano Wang, Xu Complex Variables Algebraic Geometry In an influential $L^2$ extension theorem due to Demailly, the finiteness of an $L^2$ norm called the Ohsawa norm determines whether a given holomorphic function can be extended. This result has been further generalized by Zhou and Zhu to the case when the quasi-plurisubharmonic defining function of the subvariety has non-analytic singularities. We show that, however, there exist many instances of such defining functions for which only the zero function has finite Ohsawa norm, so that the $L^2$ extension statement is void in such cases, even when it has a unique log canonical place. Such a defining function occurs already among some of the simplest non-analytic singularities, namely toric ones. |
| title | $L^2$ extension of holomorphic functions and log canonical places |
| topic | Complex Variables Algebraic Geometry |
| url | https://arxiv.org/abs/2505.00801 |