Codimension two transcendental submanifolds of projective space

Kucharz, Wojciech; Simanca, Santiago R.

doi:10.5802/aif.2561

Kucharz, Wojciech; Simanca, Santiago R.

Annales de l'Institut Fourier, Volume 60 (2010) no. 4, p. 1479-1488

Abstract
Résumé

We provide a simple characterization of codimension two submanifolds of $ℙ^{n} (ℝ)$ that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when $n \geq 6$ . If the codimension two submanifold is a nonsingular algebraic subset of $ℙ^{n} (ℝ)$ whose Zariski closure in $ℙ^{n} (ℂ)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in $ℙ^{n} (ℝ)$ .

Nous fournissons une caractérisation simple des variétés de codimension deux de $ℙ^{n} (ℝ)$ qui sont de type algébrique, et employons ce critère pour fournir des exemples des sous-variétés transcendantales quand $n \geq 6$ . Si la sous-variété de codimension deux est un sous-ensemble algébrique non singulier de $ℙ^{n} (ℝ)$ dont la fermeture de Zariski dans $ℙ^{n} (ℂ)$ est un ensemble algébrique complexe non singulier, alors ce doit être une intersection algébrique complète dans $ℙ^{n} (ℝ)$ .

MR 2722248 | Zbl 1195.14076

DOI : https://doi.org/10.5802/aif.2561
Classification: 14P25, 57R22, 57R52
Keywords: Smooth manifold, algebraic set, isotopy, complete intersection, vector bundle

@article{AIF_2010__60_4_1479_0,
     author = {Kucharz, Wojciech and Simanca, Santiago R.},
     title = {Codimension two transcendental submanifolds of projective space},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     pages = {1479-1488},
     doi = {10.5802/aif.2561},
     mrnumber = {2722248},
     zbl = {1195.14076},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_4_1479_0}
}

Kucharz, Wojciech; Simanca, Santiago R. Codimension two transcendental submanifolds of projective space. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1479-1488. doi : 10.5802/aif.2561. http://www.numdam.org/item/AIF_2010__60_4_1479_0/

References

[1] Akbulut, S.; King, H. Transcendental submanifolds of $ℝ^{n}$ , Comment. Math. Helv., Tome 68 (1993) no. 2, pp. 308-318 | Article | MR 1214234 | Zbl 0806.57017

[2] Akbulut, Selman; King, Henry Transcendental submanifolds of $ℝ ℙ^{n}$ , Comment. Math. Helv., Tome 80 (2005) no. 2, pp. 427-432 | Article | MR 2142249 | Zbl 1071.57026

[3] Bochnak, J.; Buchner, M.; Kucharz, W. Erratum: “Vector bundles over real algebraic varieties” [ $K$ -Theory 3 (1989), no. 3, p. 271–298; MR1040403 (91b:14075)], $K$ -Theory, Tome 4 (1990) no. 1, pp. 103 | MR 1040403 | Zbl 0761.14020

[4] Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 36 (1998) (Translated from the 1987 French original, Revised by the authors) | MR 1659509 | Zbl 0912.14023

[5] Davis, James F.; Kirk, Paul Lecture notes in algebraic topology, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 35 (2001) | MR 1841974 | Zbl 1018.55001

[6] Hartshorne, Robin Varieties of small codimension in projective space, Bull. Amer. Math. Soc., Tome 80 (1974), pp. 1017-1032 | Article | MR 384816 | Zbl 0304.14005

[7] Hironaka, Heisuke Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), p. 109–203; ibid. (2), Tome 79 (1964), pp. 205-326 | MR 199184 | Zbl 0122.38603

[8] Husemoller, Dale Fibre bundles, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 20 (1994) | MR 1249482 | Zbl 0307.55015

[9] King, Henry C. Approximating submanifolds of real projective space by varieties, Topology, Tome 15 (1976) no. 1, pp. 81-85 | Article | MR 396572 | Zbl 0316.57015

[10] Kucharz, Wojciech Homology classes of real algebraic sets, Ann. Inst. Fourier (Grenoble), Tome 58 (2008) no. 3, pp. 989-1022 | Article | Numdam | MR 2427517 | Zbl 1153.14035

[11] Kucharz, Wojciech Transcendental submanifolds of projective space, Comment. Math. Helv., Tome 84 (2009) no. 1, pp. 127-133 | Article | MR 2466077 | Zbl pre05508255

[12] Milnor, John W. Topology from the differentiable viewpoint, Princeton University Press, Princeton, NJ, Princeton Landmarks in Mathematics (1997) (Based on notes by David W. Weaver, Revised reprint of the 1965 original) | MR 1487640 | Zbl 1025.57002

[13] Milnor, John W.; Stasheff, James D. Characteristic classes, Princeton University Press, Princeton, N. J. (1974) (Annals of Mathematics Studies, No. 76) | MR 440554 | Zbl 0298.57008

[14] Nash, John Real algebraic manifolds, Ann. of Math. (2), Tome 56 (1952), pp. 405-421 | Article | MR 50928 | Zbl 0048.38501

[15] Steenrod, Norman The Topology of Fibre Bundles, Princeton University Press, Princeton, N. J., Princeton Mathematical Series, vol. 14 (1951) | MR 39258 | Zbl 0054.07103

[16] Tognoli, A. Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa (3), Tome 27 (1973), pp. 167-185 | Numdam | MR 396571 | Zbl 0263.57011