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

The goal of this paper is to understand the properties of meromorphic mappings with values in two model complex Hibert manifolds: projective Hilbert space $\text{pp}(l^2)$ and Sobolev loop space of the Riemann sphere $L{\text{pp}}^1$. It occurs that these properties are quite different. Based on our study we obtain as a corollary that $L{\text{pp}}^1$ does not admit a closed holomorphic embedding to $\text{pp}(l^2)$. In other words $L{\text{pp}}^1$ 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 $L{\text{pp}}^1$ doesn’t admit even a non-degenerate meromorphic map to $\text{pp}(l^2)$.