Rational approximation of holomorphic maps

Bochnak, Jacek; Kucharz, Wojciech

doi:10.5802/aif.3542

Rational approximation of holomorphic maps
[Approximation rationnelle des applications holomorphes]

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, Tome 73 (2023) no. 3, pp. 1115-1131.

Résumé
Abstract

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 $\tilde{φ} |_{K} = φ$ .

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 $\tilde{φ} |_{K} = φ$ .

Reçu le : 2020-12-07
Révisé le : 2021-11-22
Accepté le : 2021-12-01
Publié le : 2023-05-12

DOI : 10.5802/aif.3542

Classification : 32Q56, 41A20, 14E05, 14M17
Keywords: Algebraic manifold, holomorphic map, regular map, approximation.
Mot clés : Variété algébrique, application holomorphe, application régulière, approximation.

Affiliations des auteurs :

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)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Rational approximation of holomorphic maps},
     journal = {Annales de l'Institut Fourier},
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Bochnak, Jacek; Kucharz, Wojciech. Rational approximation of holomorphic maps. Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1115-1131. doi : 10.5802/aif.3542. https://aif.centre-mersenne.org/articles/10.5802/aif.3542/

Bibliographie
Cité par

[1] Arzhantsev, I.; Flenner, H.; Kaliman, S.; Kutzschebauch, F.; Zaidenberg, M. Flexible varieties and automorphism groups, Duke Math. J., Volume 162 (2013) no. 4, pp. 767-823 | DOI | MR | Zbl

[2] Atiyah, M. F.; Hirzebruch, F. Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math., Volume III, American Mathematical Society (1961), pp. 7-38 | DOI | MR | Zbl

[3] Benoist, Olivier; Wittenberg, Olivier The tight approximation property, J. Reine Angew. Math., Volume 776 (2021), pp. 151-200 | DOI | MR | Zbl

[4] Bochnak, J.; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Ergeb. Math. Grenzgeb., 3. Folge, 36, Springer-Verlag, Berlin, 1998, x+430 pages | DOI | MR | Zbl

[5] Bochnak, J.; Kucharz, W. Realization of homotopy classes by algebraic mappings, J. Reine Angew. Math., Volume 377 (1987), pp. 159-169 | MR | Zbl

[6] Bochnak, J.; Kucharz, W. Approximation of holomorphic maps by algebraic morphisms, Ann. Polon. Math., Volume 80 (2003), pp. 85-92 | DOI | MR | Zbl

[7] Bredon, Glen E. Topology and geometry, Graduate Texts in Mathematics, 139, Springer-Verlag, New York, 1993, xiv+557 pages | DOI | MR | Zbl

[8] Bröcker, Theodor; Jänich, Klaus Introduction to differential topology, Cambridge University Press, Cambridge-New York, 1982, vii+160 pages (Translated from the German by C. B. Thomas and M. J. Thomas) | MR | Zbl

[9] Chevalley, C. On algebraic group varieties, J. Math. Soc. Japan, Volume 6 (1954), pp. 303-324 | DOI | MR | Zbl

[10] Demailly, Jean-Pierre; Lempert, László; Shiffman, Bernard Algebraic approximations of holomorphic maps from Stein domains to projective manifolds, Duke Math. J., Volume 76 (1994) no. 2, pp. 333-363 | DOI | MR | Zbl

[11] Fornæss, John Erik; Forstnerič, Franc; Wold, Erlend F. Holomorphic approximation: the legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan, Advancements in complex analysis – from theory to practice, Springer, Cham, 2020, pp. 133-192 | DOI | MR | Zbl

[12] Forstnerič, Franc Holomorphic flexibility properties of complex manifolds, Amer. J. Math., Volume 128 (2006) no. 1, pp. 239-270 | DOI | MR | Zbl

[13] Forstnerič, Franc Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb., 3. Folge, 56, Springer, Cham, 2017, xiv+562 pages | DOI | MR | Zbl

[14] Forstnerič, Franc Developments in Oka theory since 2017 (2020) (https://arxiv.org/abs/2006.07888)

[15] Fulton, William Intersection theory, Ergeb. Math. Grenzgeb., 3. Folge, 2, Springer-Verlag, Berlin, 1984, xi+470 pages | DOI | MR | Zbl

[16] Grauert, Hans Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., Volume 135 (1958), pp. 263-273 | DOI | MR | Zbl

[17] Grauert, Hans On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2), Volume 68 (1958), pp. 460-472 | DOI | MR | Zbl

[18] Gromov, M. Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., Volume 2 (1989) no. 4, pp. 851-897 | DOI | MR | Zbl

[19] Gunning, Robert C.; Rossi, Hugo Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965, xiv+317 pages | MR | Zbl

[20] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990, xii+254 pages | MR | Zbl

[21] Husemoller, Dale Fibre bundles, Graduate Texts in Mathematics, 20, Springer-Verlag, New York, 1994, xx+353 pages | DOI | MR | Zbl

[22] Kaliman, Shulim; Kutzschebauch, Frank; Truong, Tuyen Trung On subelliptic manifolds, Israel J. Math., Volume 228 (2018) no. 1, pp. 229-247 | DOI | MR | Zbl

[23] Kucharz, W. The Runge approximation problem for holomorphic maps into Grassmannians, Math. Z., Volume 218 (1995) no. 3, pp. 343-348 | DOI | MR | Zbl

[24] Kusakabe, Yuta An implicit function theorem for sprays and applications to Oka theory, Internat. J. Math., Volume 31 (2020) no. 9, 2050071, 9 pages | DOI | MR | Zbl

[25] Lárusson, Finnur; Truong, Tuyen Trung Approximation and interpolation of regular maps from affine varieties to algebraic manifolds, Math. Scand., Volume 125 (2019) no. 2, pp. 199-209 | DOI | MR | Zbl

[26] Lempert, László Algebraic approximations in analytic geometry, Invent. Math., Volume 121 (1995) no. 2, pp. 335-353 | DOI | MR | Zbl

[27] Serre, Jean-Pierre Faisceaux algébriques cohérents, Ann. of Math. (2), Volume 61 (1955), pp. 197-278 | DOI | MR | Zbl

[28] Whitney, Hassler Differentiable manifolds, Ann. of Math. (2), Volume 37 (1936) no. 3, pp. 645-680 | DOI | MR | Zbl

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