Let be a complex nonsingular affine algebraic variety, a compact holomorphically convex subset of , and a homogeneous complex manifold for some complex linear algebraic group. We prove that a holomorphic map can be uniformly approximated on by regular maps if and only if is homotopic to a regular map . However, it may happen that a null homotopic holomorphic map does not admit uniform approximation on by regular maps . Here, a map is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood of and a holomorphic (resp. regular) map such that .
Soit une variété algébrique affine non singulière complexe, soit un sous-ensemble holomorphiquement convexe de , et soit une variété algébrique complexe homogène pour un groupe algébrique linéaire complexe. Nous montrons qu’une application holomorphe peut être uniformément approchée sur par des applications régulières si et seulement si est homotope à une application régulière . Cependant, il peut arriver qu’une application holomorphe qui est null-homotope n’admette pas d’approximation uniforme par des applications régulières . Ici, on dit qu’une application est holomorphe (resp. régulière) s’il existe un voisinage ouvert (resp. ouvert de Zariski) de et une application holomorphe (resp. régulière) telle que .
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Keywords: Algebraic manifold, holomorphic map, regular map, approximation.
Mot clés : Variété algébrique, application holomorphe, application régulière, approximation.
Bochnak, Jacek 1; Kucharz, Wojciech 2
@article{AIF_2023__73_3_1115_0, author = {Bochnak, Jacek and Kucharz, Wojciech}, title = {Rational approximation of holomorphic maps}, journal = {Annales de l'Institut Fourier}, pages = {1115--1131}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {3}, year = {2023}, doi = {10.5802/aif.3542}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3542/} }
TY - JOUR AU - Bochnak, Jacek AU - Kucharz, Wojciech TI - Rational approximation of holomorphic maps JO - Annales de l'Institut Fourier PY - 2023 SP - 1115 EP - 1131 VL - 73 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3542/ DO - 10.5802/aif.3542 LA - en ID - AIF_2023__73_3_1115_0 ER -
%0 Journal Article %A Bochnak, Jacek %A Kucharz, Wojciech %T Rational approximation of holomorphic maps %J Annales de l'Institut Fourier %D 2023 %P 1115-1131 %V 73 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3542/ %R 10.5802/aif.3542 %G en %F AIF_2023__73_3_1115_0
Bochnak, Jacek; Kucharz, Wojciech. Rational approximation of holomorphic maps. Annales de l'Institut Fourier, Volume 73 (2023) no. 3, pp. 1115-1131. doi : 10.5802/aif.3542. https://aif.centre-mersenne.org/articles/10.5802/aif.3542/
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