Codimension two transcendental submanifolds of projective space
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1479-1488.

We provide a simple characterization of codimension two submanifolds of n () that are of algebraic type, and use this criterion to provide examples of transcendental submanifolds when n6. If the codimension two submanifold is a nonsingular algebraic subset of n () whose Zariski closure in n () is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in n ().

Nous fournissons une caractérisation simple des variétés de codimension deux de n () qui sont de type algébrique, et employons ce critère pour fournir des exemples des sous-variétés transcendantales quand n6. Si la sous-variété de codimension deux est un sous-ensemble algébrique non singulier de n () dont la fermeture de Zariski dans n () est un ensemble algébrique complexe non singulier, alors ce doit être une intersection algébrique complète dans n ().

DOI: 10.5802/aif.2561
Classification: 14P25, 57R22, 57R52
Keywords: Smooth manifold, algebraic set, isotopy, complete intersection, vector bundle
Mot clés : variétés différentiables, ensemble algébrique, isotopie, intersection complète, fibré vectoriel
Kucharz, Wojciech 1; Simanca, Santiago R. 2

1 Jagiellonian University Institute of Mathematics Lojasiewicza 6 30-348 Krakow (Poland)
2 University of New Mexico Department of Mathematics & Statistics Albuquerque, NM 87131 (USA)
@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{\textquoteright}institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     doi = {10.5802/aif.2561},
     zbl = {1195.14076},
     mrnumber = {2722248},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2561/}
}
TY  - JOUR
AU  - Kucharz, Wojciech
AU  - Simanca, Santiago R.
TI  - Codimension two transcendental submanifolds of projective space
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 1479
EP  - 1488
VL  - 60
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2561/
DO  - 10.5802/aif.2561
LA  - en
ID  - AIF_2010__60_4_1479_0
ER  - 
%0 Journal Article
%A Kucharz, Wojciech
%A Simanca, Santiago R.
%T Codimension two transcendental submanifolds of projective space
%J Annales de l'Institut Fourier
%D 2010
%P 1479-1488
%V 60
%N 4
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2561/
%R 10.5802/aif.2561
%G en
%F 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://archive.numdam.org/articles/10.5802/aif.2561/

[1] Akbulut, S.; King, H. Transcendental submanifolds of n , Comment. Math. Helv., Volume 68 (1993) no. 2, pp. 308-318 | DOI | MR | Zbl

[2] Akbulut, Selman; King, Henry Transcendental submanifolds of n , Comment. Math. Helv., Volume 80 (2005) no. 2, pp. 427-432 | DOI | MR | Zbl

[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, Volume 4 (1990) no. 1, pp. 103 | MR | Zbl

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

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

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

[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), Volume 79 (1964), pp. 205-326 | MR | Zbl

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

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

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

[11] Kucharz, Wojciech Transcendental submanifolds of projective space, Comment. Math. Helv., Volume 84 (2009) no. 1, pp. 127-133 | DOI | MR

[12] Milnor, John W. 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 | Zbl

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

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

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

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

Cited by Sources: