Holomorphic self-maps of the loop space of $\mathbb{P}^n$

TL;DR

This paper characterizes a class of holomorphic self-maps, including automorphisms, on the infinite-dimensional complex manifold formed by loop spaces of complex projective spaces, enriching understanding of their structure.

Contribution

It identifies and describes a specific class of holomorphic self-maps of the loop space of b^n, including all automorphisms, advancing the theory of infinite-dimensional complex manifolds.

Findings

Identified a class of holomorphic self-maps of the loop space.

Included all automorphisms within this class.

Enhanced understanding of the structure of loop space automorphisms.

Abstract

The loop space $L\mathbb{P}^n$ of the complex projective space $\mathbb{P}^n$ consisting of all $C^k$ or Sobolev $W^{k, \, p}$ maps $S^1 \to \mathbb{P}^n$ is an infinite dimensional complex manifold. We identify a class of holomorphic self-maps of $L\mathbb{P}^n$, including all automorphisms.