Rational approximation of holomorphic maps

TL;DR

This paper establishes conditions under which holomorphic maps from certain complex varieties to homogeneous varieties can be uniformly approximated by regular maps, linking approximation to homotopy classes.

Contribution

It proves that homotopy equivalence to a regular map characterizes uniform approximation of holomorphic maps by regular maps.

Findings

Holomorphic maps homotopic to regular maps can be uniformly approximated.

Null homotopic holomorphic maps may not be approximable by regular maps.

Approximation depends on the homotopy class of the map.

Abstract

Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety 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 can happen that a null homotopic holomorphic map K-->Y does not admit uniform approximation on K by regular maps X-->Y. Here, a map g:K-->Y is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood U of K in X and a holomorphic (resp. regular) map h:U-->Y such that h|K=g.