Rational approximation of holomorphic maps

Bochnak, Jacek; Kucharz, Wojciech

doi:10.5802/aif.3542

Bochnak, Jacek ¹; Kucharz, Wojciech ²

¹ Le Pont de l’Étang 8 1323 Romainmôtier (Switzerland)
² Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków (Poland)

Annales de l'Institut Fourier, Online first, 17 p.

Abstract
Résumé

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 $f : K \to Y$ can be uniformly approximated on $K$ by regular maps $K \to Y$ if and only if $f$ is homotopic to a regular map $K \to Y$ . However, it may happen that a null homotopic holomorphic map $K \to Y$ does not admit uniform approximation on $K$ by regular maps $X \to Y$ . Here, a map $φ : K \to Y$ is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood $U \subseteq X$ of $K$ and a holomorphic (resp. regular) map $\tilde{φ} : U \to Y$ such that $\tilde{φ} |_{K} = φ$ .

Soit $X$ une variété algébrique affine non singulière complexe, soit $K$ un sous-ensemble holomorphiquement convexe de $X$ , et soit $Y$ une variété algébrique complexe homogène pour un groupe algébrique linéaire complexe. Nous montrons qu’une application holomorphe $f : K \to Y$ peut être uniformément approchée sur $K$ par des applications régulières $K \to Y$ si et seulement si $f$ est homotope à une application régulière $K \to Y$ . Cependant, il peut arriver qu’une application holomorphe $K \to Y$ qui est null-homotope n’admette pas d’approximation uniforme par des applications régulières $X \to Y$ . Ici, on dit qu’une application $φ : K \to Y$ est holomorphe (resp. régulière) s’il existe un voisinage ouvert (resp. ouvert de Zariski) $U \subset X$ de $K$ et une application holomorphe (resp. régulière) $\tilde{φ} : U \to Y$ telle que $\tilde{φ} |_{K} = φ$ .

Received: 2020-12-07
Revised: 2021-11-22
Accepted: 2021-12-01
Online First: 2022-11-28

DOI: 10.5802/aif.3542

Classification: 32Q56, 41A20, 14E05, 14M17
Keywords: Algebraic manifold, holomorphic map, regular map, approximation.

Author's affiliations:

Bochnak, Jacek ¹; Kucharz, Wojciech ²

¹ Le Pont de l’Étang 8 1323 Romainmôtier (Switzerland)
² Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków (Poland)

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     title = {Rational approximation of holomorphic maps},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3542},
     language = {en},
     note = {Online first},
}

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Bochnak, Jacek; Kucharz, Wojciech. Rational approximation of holomorphic maps. Annales de l'Institut Fourier, Online first, 17 p.

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