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.
@unpublished{AIF_0__0_0_A119_0, author = {Bochnak, Jacek and Kucharz, Wojciech}, title = {Rational approximation of holomorphic maps}, journal = {Annales de l'Institut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}, }
TY - UNPB AU - Bochnak, Jacek AU - Kucharz, Wojciech TI - Rational approximation of holomorphic maps JO - Annales de l'Institut Fourier PY - 2022 DA - 2022/// PB - Association des Annales de l’institut Fourier N1 - Online first UR - https://doi.org/10.5802/aif.3542 DO - 10.5802/aif.3542 LA - en ID - AIF_0__0_0_A119_0 ER -
Bochnak, Jacek; Kucharz, Wojciech. Rational approximation of holomorphic maps. Annales de l'Institut Fourier, Online first, 17 p.
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