Rational approximation of holomorphic maps

Abstract

Let X be a complex nonsingular affine algebraic variety, K a compact holomorphically convex subset of X, and Y a homogeneous complex manifold for some complex linear algebraic group. We prove that a holomorphic map f:K→Y can be uniformly approximated on K by regular maps K→Y if and only if f is homotopic to a regular map K→Y. However, it may happen that a null homotopic holomorphic map K→Y does not admit uniform approximation on K by regular maps X→Y. Here, a map φ:K→Y is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood U⊆X of K and a holomorphic (resp. regular) map φ ˜:U→Y such that φ ˜| K =φ.