<p>The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {P}(l^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and Sobolev loop space of the Riemann sphere <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. It occurs that these properties are quite different. Based on our study we obtain as a corollary that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is not biholomorphic to a submanifold of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {P}(l^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In other words <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a Hilbert projective variety despite of the fact that it is Kähler and meromorphic functions separate points on it. Moreover, we prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> doesn’t admit even a non-degenerate meromorphic map to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {P}(l^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">(</mo> <msup> <mi>l</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Non-Existence of a Holomorphic Embedding of the Sobolev Loop Space into the Projective Hilbert Space

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发表日期：

2026年7月16日

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M. Anakkar,
S. Ivashkovych

摘要

The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space $\mathbb {P}(l^2)$ $P (l^{2})$ and Sobolev loop space of the Riemann sphere $L\mathbb {P}^1$ $L P^{1}$ . It occurs that these properties are quite different. Based on our study we obtain as a corollary that $L\mathbb {P}^1$ $L P^{1}$ is not biholomorphic to a submanifold of $\mathbb {P}(l^2)$ $P (l^{2})$ . In other words $L\mathbb {P}^1$ $L P^{1}$ is a Hilbert projective variety despite of the fact that it is Kähler and meromorphic functions separate points on it. Moreover, we prove that $L\mathbb {P}^1$ $L P^{1}$ doesn’t admit even a non-degenerate meromorphic map to $\mathbb {P}(l^2)$ $P (l^{2})$ .