Let 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 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.
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@article{CRMATH_2005__340_12_889_0, author = {Samokhin, Alexander}, title = {On the derived category of coherent sheaves on a 5-dimensional {Fano} variety}, journal = {Comptes Rendus. Math\'ematique}, pages = {889--893}, publisher = {Elsevier}, volume = {340}, number = {12}, year = {2005}, doi = {10.1016/j.crma.2005.04.033}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.04.033/} }
TY - JOUR AU - Samokhin, Alexander TI - On the derived category of coherent sheaves on a 5-dimensional Fano variety JO - Comptes Rendus. Mathématique PY - 2005 SP - 889 EP - 893 VL - 340 IS - 12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.04.033/ DO - 10.1016/j.crma.2005.04.033 LA - en ID - CRMATH_2005__340_12_889_0 ER -
%0 Journal Article %A Samokhin, Alexander %T On the derived category of coherent sheaves on a 5-dimensional Fano variety %J Comptes Rendus. Mathématique %D 2005 %P 889-893 %V 340 %N 12 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2005.04.033/ %R 10.1016/j.crma.2005.04.033 %G en %F CRMATH_2005__340_12_889_0
Samokhin, Alexander. 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. http://archive.numdam.org/articles/10.1016/j.crma.2005.04.033/
[1] Coherent sheaves on and problems of linear algebra, Funk. Anal., Volume 12 (1978), pp. 68-69
[2] Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk USSR Ser. Mat., Volume 53 (1989) no. 1, pp. 23-42
[3] Representable functors, Serre functors, and mutations, Izv. Akad. Nauk USSR Ser. Mat., Volume 53 (1989) no. 6, pp. 519-541
[4] On the derived category of coherent sheaves on some homogeneous spaces, Invent. Math., Volume 92 (1988), pp. 479-508
[5] 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 , preprint MPI, 24, 1997
[7] A. Kuznetsov, in preparation
[8] Exceptional set of vector bundles on the variety , Vestnik Moskov. Univ. Ser. I Mat. Mekh., Volume 5 (1991), pp. 69-71
[9] The derived category of coherent sheaves on , Uspekhi Math. Nauk, Volume 56 (2001) no. 3, pp. 592-594
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⁎ This work was supported in part by the French Government fellowship and by the RFFI award No 02-01-22005.