The Picard group of the loop space of the Riemann sphere

Abstract

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbeault group. The group of Mobius transformations $G$ and its loop group $LG$ act on this loop space. We prove that an element of the Picard group is $LG$-fixed if it is $G$-fixed; thus completely answer the question by Millson and Zombro about $G$-equivariant projective embedding of the loop space of the Riemann sphere.

for other preprints of Ning Zhang see http://arxiv.org/a/zhang_n_1

References in corpus (1)

• Holomorphic line bundles on the loop space of the Riemann sphere

Cited by in corpus (2)

• Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces
• Holomorphic self-maps of the loop space of ${\mathbb{P}}^n$