Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group $ \mathbb{Z}$

Nicolas Perrin

doi:10.1016/j.crma.2007.06.012

Algebraic Geometry

Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group

Z

Presented by: Jean-Pierre Demailly

Nicolas Perrin ¹

¹ Université Pierre-et-Marie-Curie - Paris 6, institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 155-160.

Abstract
Résumé

We investigate a method proposed by E. Arrondo and J. Caravantes to study the Picard group of a smooth low-codimensional subvariety X in a variety Y when Y is homogeneous. We prove that this method is strongly related to the signature $σ_{Y}$ of the Poincaré pairing on the middle cohomology of Y. We give under some topological assumptions a bound on the rank of Picard group $Pic (X)$ in terms of $σ_{Y}$ and remove these assumptions for Grassmannians to recover the main result of E. Arrondo and J. Caravantes.

E. Arrondo et J. Caravantes ont proposé une méthode pour étudier le groupe de Picard $Pic (X)$ d'une sous-variété lisse X d'une variété Y. Dans le cas où Y est homogène, nous montrons que cette méthode est intimement liée à la signature $σ_{Y}$ de l'accouplement de Poincaré sur la cohomologie de dimension moitié de Y. Nous donnons, sous certaines hypothèses topologiques, une borne sur le rang de $Pic (X)$ en fonction de $σ_{Y}$ . Dans le cas des grassmanniennes, ces conditions topologiques sont satisfaites et nous obtenons une généralisation des résultats de E. Arrondo et J. Caravantes.

Received: 2007-02-20
Accepted: 2007-06-12
Published online: 2007-07-20

DOI: 10.1016/j.crma.2007.06.012

Author's affiliations:

Nicolas Perrin ¹

¹ Université Pierre-et-Marie-Curie - Paris 6, institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France

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Nicolas Perrin. Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group $ \mathbb{Z}$. Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 155-160. doi : 10.1016/j.crma.2007.06.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.012/

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