Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908657865719808 |
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| author | M., Anakkar Ivashkovich, S. |
| author_facet | M., Anakkar Ivashkovich, S. |
| contents | The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space $\pp(l^2)$ and Sobolev loop space of the Riemann sphere $L\pp^1$. It occurs that these properties are quite different. Based on our study we obtain as a corollary that $L\pp^1$ does not admit a closed holomorphic embedding to $\pp(l^2)$. In other words $L\pp^1$ is {\slsf not} a projective Hilbert variety despite of the fact that it is Kähler and meromorphic functions separate points on it. Moreover, we prove that $L\pp^1$ doesn't admit even a non-degenerate meromorphic map to $\pp (l^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space M., Anakkar Ivashkovich, S. Complex Variables Primary - 32D15, Secondary - 32A20, 46G20, 46T25 The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space $\pp(l^2)$ and Sobolev loop space of the Riemann sphere $L\pp^1$. It occurs that these properties are quite different. Based on our study we obtain as a corollary that $L\pp^1$ does not admit a closed holomorphic embedding to $\pp(l^2)$. In other words $L\pp^1$ is {\slsf not} a projective Hilbert variety despite of the fact that it is Kähler and meromorphic functions separate points on it. Moreover, we prove that $L\pp^1$ doesn't admit even a non-degenerate meromorphic map to $\pp (l^2)$. |
| title | Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space |
| topic | Complex Variables Primary - 32D15, Secondary - 32A20, 46G20, 46T25 |
| url | https://arxiv.org/abs/2406.01404 |