Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space

Anakkar M.; S. Ivashkovich

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arXiv · 2406.01404

Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space

Anakkar M.·S. Ivashkovich

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Abstract

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 K\"ahler and meromorphic functions separate points on it. Moreover, we prove that $L\pp^1$ doesn't admit even a non-degenerate meromorphic map to $\pp (l^2)$.

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BibTeX RIS

Anakkar M., S. Ivashkovich. 2024-06-03. Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space. https://arxiv.org/abs/2406.01404

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