On the derived category of coherent sheaves on a 5-dimensional Fano variety

Alexander Samokhin

doi:10.1016/j.crma.2005.04.033

Algebraic Geometry

On the derived category of coherent sheaves on a 5-dimensional Fano variety
[Sur la catégorie dérivée de faisceaux cohérents sur une variété de Fano de dimension 5]

Présenté par : Bernard Malgrange

Alexander Samokhin ¹

¹ Département de mathématiques, LAGA (UMR 7539), institut Galilée, université Paris 13, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 889-893.

Abstract (VO)
Résumé

Let ${LG}_{3}^{C}$ be the Grassmannian of Lagrangian planes in a six-dimensional vector space V. It is a six-dimensional Fano variety of index 4. Consider its smooth hyperplane section. We show that in the derived category of coherent sheaves on such a hyperplane section there exists an exceptional collection, generating the derived category.

Soit ${LG}_{3}^{C}$ la grassmannienne des plans lagrangiens dans un espace vectoriel V de dimension 6. C'est une variété de Fano d'indice 4. Considérons sa section lisse par un hyperplan. Nous montrons que dans la catégorie dérivée des faisceaux cohérents sur une telle section il existe une collection exceptionnelle qui engendre la catégorie dérivée.

Reçu le : 2004-05-10
Accepté le : 2005-04-18
Publié le : 2005-05-24

DOI : 10.1016/j.crma.2005.04.033

Affiliations des auteurs :

Alexander Samokhin ¹

¹ Département de mathématiques, LAGA (UMR 7539), institut Galilée, université Paris 13, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

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Alexander Samokhin. On the derived category of coherent sheaves on a 5-dimensional Fano variety. Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 889-893. doi : 10.1016/j.crma.2005.04.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.033/

Bibliographie
Cité par

[1] A. Beilinson Coherent sheaves on $P^{n}$ and problems of linear algebra, Funk. Anal., Volume 12 (1978), pp. 68-69

[2] A. Bondal Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk USSR Ser. Mat., Volume 53 (1989) no. 1, pp. 23-42

[3] A. Bondal; M. Kapranov Representable functors, Serre functors, and mutations, Izv. Akad. Nauk USSR Ser. Mat., Volume 53 (1989) no. 6, pp. 519-541

[4] M. Kapranov On the derived category of coherent sheaves on some homogeneous spaces, Invent. Math., Volume 92 (1988), pp. 479-508

[5] M. Kontsevich Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 120-139

[6] A. Kuznetsov, Fano threefolds $V_{22}$ , preprint MPI, 24, 1997

[7] A. Kuznetsov, in preparation

[8] D. Orlov Exceptional set of vector bundles on the variety $V_{5}$ , Vestnik Moskov. Univ. Ser. I Mat. Mekh., Volume 5 (1991), pp. 69-71

[9] A. Samokhin The derived category of coherent sheaves on ${LG}_{3}^{C}$ , Uspekhi Math. Nauk, Volume 56 (2001) no. 3, pp. 592-594

Cité par Sources :

^⁎ This work was supported in part by the French Government fellowship and by the RFFI award No 02-01-22005.

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