Codimension two transcendental submanifolds of projective space

[Sous-variétés transcendantales de codimension deux dans l’espace projectif]

Kucharz, Wojciech ; Simanca, Santiago R.

Annales de l'Institut Fourier, Tome 60 (2010), p. 1479-1488 / Harvested from Numdam

Résumé

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

Abstract

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

Détail

Publié le : 2010-01-01

MR 2722248 | Zbl 1195.14076

DOI : https://doi.org/10.5802/aif.2561
Classification:  14P25,  57R22,  57R52
Mots clés: variétés différentiables, ensemble algébrique, isotopie, intersection complète, fibré vectoriel

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