Nous fournissons une caractérisation simple des variétés de codimension deux de qui sont de type algébrique, et employons ce critère pour fournir des exemples des sous-variétés transcendantales quand . Si la sous-variété de codimension deux est un sous-ensemble algébrique non singulier de dont la fermeture de Zariski dans est un ensemble algébrique complexe non singulier, alors ce doit être une intersection algébrique complète dans .
We provide a simple characterization of codimension two submanifolds of that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when . If the codimension two submanifold is a nonsingular algebraic subset of whose Zariski closure in is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in .
Classification : 14P25, 57R22, 57R52
Mots clés : variétés différentiables, ensemble algébrique, isotopie, intersection complète, fibré vectoriel
@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}, pages = {1479--1488}, publisher = {Association des Annales de l'institut Fourier}, volume = {60}, number = {4}, year = {2010}, doi = {10.5802/aif.2561}, mrnumber = {2722248}, zbl = {1195.14076}, language = {en}, url = {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, Tome 60 (2010) no. 4, pp. 1479-1488. doi : 10.5802/aif.2561. http://www.numdam.org/item/AIF_2010__60_4_1479_0/
[1] Transcendental submanifolds of , Comment. Math. Helv., Volume 68 (1993) no. 2, pp. 308-318 | Article | MR 1214234 | Zbl 0806.57017
[2] Transcendental submanifolds of , Comment. Math. Helv., Volume 80 (2005) no. 2, pp. 427-432 | Article | MR 2142249 | Zbl 1071.57026
[3] Erratum: “Vector bundles over real algebraic varieties” [-Theory 3 (1989), no. 3, p. 271–298; MR1040403 (91b:14075)], -Theory, Volume 4 (1990) no. 1, pp. 103 | MR 1040403 | Zbl 0761.14020
[4] Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 36, Springer-Verlag, Berlin, 1998 (Translated from the 1987 French original, Revised by the authors) | MR 1659509 | Zbl 0912.14023
[5] Lecture notes in algebraic topology, Graduate Studies in Mathematics, Volume 35, American Mathematical Society, Providence, RI, 2001 | MR 1841974 | Zbl 1018.55001
[6] Varieties of small codimension in projective space, Bull. Amer. Math. Soc., Volume 80 (1974), pp. 1017-1032 | Article | MR 384816 | Zbl 0304.14005
[7] 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), Volume 79 (1964), pp. 205-326 | MR 199184 | Zbl 0122.38603
[8] Fibre bundles, Graduate Texts in Mathematics, Volume 20, Springer-Verlag, New York, 1994 | MR 1249482 | Zbl 0307.55015
[9] Approximating submanifolds of real projective space by varieties, Topology, Volume 15 (1976) no. 1, pp. 81-85 | Article | MR 396572 | Zbl 0316.57015
[10] Homology classes of real algebraic sets, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 3, pp. 989-1022 | Article | Numdam | MR 2427517 | Zbl 1153.14035
[11] Transcendental submanifolds of projective space, Comment. Math. Helv., Volume 84 (2009) no. 1, pp. 127-133 | Article | MR 2466077 | Zbl pre05508255
[12] Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997 (Based on notes by David W. Weaver, Revised reprint of the 1965 original) | MR 1487640 | Zbl 1025.57002
[13] Characteristic classes, Princeton University Press, Princeton, N. J., 1974 (Annals of Mathematics Studies, No. 76) | MR 440554 | Zbl 0298.57008
[14] Real algebraic manifolds, Ann. of Math. (2), Volume 56 (1952), pp. 405-421 | Article | MR 50928 | Zbl 0048.38501
[15] The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951 | MR 39258 | Zbl 0054.07103
[16] Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa (3), Volume 27 (1973), pp. 167-185 | Numdam | MR 396571 | Zbl 0263.57011