Rational approximation of holomorphic maps

2020/12/04 by Bochnak, Jacek, Kucharz, Wojciech
#14E05 #14M17 #32Q56 #41A20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.02562

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.

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