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

TL;DR

This paper proves that the Sobolev loop space of the Riemann sphere cannot be holomorphically embedded into the projective Hilbert space, highlighting fundamental differences in their complex geometric properties.

Contribution

It establishes the non-existence of holomorphic and non-degenerate meromorphic embeddings of the Sobolev loop space into the projective Hilbert space.

Findings

Lb^1 is not a projective Hilbert variety

No holomorphic embedding of Lb^1 into bp(l^2) exists

Lb^1 does not admit a non-degenerate meromorphic map to bp(l^2)

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 $\pp (l^2)$.