Codimension two transcendental submanifolds of projective space

Kucharz, Wojciech; Simanca, Santiago R.

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

Nous fournissons une caractérisation simple des variétés de codimension deux de ${\mathbb{P}}^n\left(ℝ\right)$ 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 ${\mathbb{P}}^n\left(ℝ\right)$ dont la fermeture de Zariski dans ${\mathbb{P}}^n\left(ℂ\right)$ est un ensemble algébrique complexe non singulier, alors ce doit être une intersection algébrique complète dans ${\mathbb{P}}^n\left(ℝ\right)$.

We provide a simple characterization of codimension two submanifolds of ${\mathbb{P}}^n\left(ℝ\right)$ 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 ${\mathbb{P}}^n\left(ℝ\right)$ whose Zariski closure in ${\mathbb{P}}^n\left(ℂ\right)$ is a nonsingular complex algebraic set, then it must be an algebraic complete intersection in ${\mathbb{P}}^n\left(ℝ\right)$.

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

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