Holomorphic Functions on (Generalised) Loop Spaces

Abstract

Let ℙ n denote n dimensional complex projective space. Given k = 0,1,..., ∞, a map χ0 : S¹ -> ℙ n of class Ck, and a closed subset A ⊂ S¹ consider the space Y of those C k maps χ : S¹ -> ℙ n that agree with χ₀ along A . The space Y can be endowed with the structure of an infinite dimensional complex manifold, and our main result identifies the space of holomorphic functions on Y.