Rational approximation of holomorphic maps
Annales de l'Institut Fourier, Online first, 17 p.

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:KY can be uniformly approximated on K by regular maps KY if and only if f is homotopic to a regular map KY. However, it may happen that a null homotopic holomorphic map KY does not admit uniform approximation on K by regular maps XY. Here, a map φ:KY is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood UX of K and a holomorphic (resp. regular) map φ ˜:UY such that 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:KY peut être uniformément approchée sur K par des applications régulières KY si et seulement si f est homotope à une application régulière KY. Cependant, il peut arriver qu’une application holomorphe KY qui est null-homotope n’admette pas d’approximation uniforme par des applications régulières XY. Ici, on dit qu’une application φ:KY est holomorphe (resp. régulière) s’il existe un voisinage ouvert (resp. ouvert de Zariski) UX de K et une application holomorphe (resp. régulière) φ ˜:UY telle que φ ˜| K =φ.

Online First:
DOI: 10.5802/aif.3542
Classification: 32Q56,  41A20,  14E05,  14M17
Keywords: Algebraic manifold, holomorphic map, regular map, approximation.
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)
     author = {Bochnak, Jacek and Kucharz, Wojciech},
     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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