Lefschetz for Local Picard groups
[Lefschetz pour les groupes de Picard locaux]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 4, pp. 833-849.

Nous prouvons un renforcement du théorème de l'hyperplan de Grothendieck-Lefschetz pour les groupes locaux de Picard conjecturés par Kollár. Notre approche, qui s'appuie sur des résultats en fermetures absolues, conduit également à un théorème de restriction pour les faisceaux de rang supérieur sur les variétés projectives en caractéristique positive.

We prove a strengthening of the Grothendieck-Lefschetz hyperplane theorem for local Picard groups conjectured by Kollár. Our approach, which relies on acyclicity results for absolute integral closures, also leads to a restriction theorem for higher rank bundles on projective varieties in positive characteristic.

Publié le :
DOI : 10.24033/asens.2228
Classification : 14B05, 14B15, 14C22, 13A35
Keywords: Picard groups, local rings, absolute integral closure, vanishing theorems, Grothendieck-Lefschetz
Mot clés : Groupes de Picard, anneaux locaux, fermeture intégrale absolue, Grothendieck-Lefschetz. 
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     author = {Bhatt, Bhargav and de Jong, Aise Johan},
     title = {Lefschetz for {Local} {Picard} groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {833--849},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
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Bhatt, Bhargav; de Jong, Aise Johan. Lefschetz for Local Picard groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 4, pp. 833-849. doi : 10.24033/asens.2228. http://www.numdam.org/articles/10.24033/asens.2228/

Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4) (Artin, M.; Grothendieck, A.; Verdier, J.-L., eds.), Lecture Notes in Math., 305, Springer, 1973, 640 pages | MR | Zbl

Antei, M.; Mehta, V. B. Vector bundles over normal varieties trivialized by finite morphisms, Arch. Math. (Basel), Volume 97 (2011), pp. 523-527 (ISSN: 0003-889X) | DOI | MR | Zbl

Artin, M. On the joins of Hensel rings, Advances in Math., Volume 7 (1971), pp. 282-296 (ISSN: 0001-8708) | DOI | MR | Zbl

Conrad, B.; de Jong, A. J. Approximation of versal deformations, J. Algebra, Volume 255 (2002), pp. 489-515 (ISSN: 0021-8693) | DOI | MR | Zbl

Fujita, T., Algebraic geometry (Tokyo/Kyoto, 1982) (Lecture Notes in Math.), Volume 1016, Springer, 1983, pp. 519-528 | DOI | MR | Zbl

Grothendieck, A., Documents Mathématiques (Paris), 4, Soc. Math. France, 2005, 208 pages (ISBN: 2-85629-169-4) | MR | Zbl

Hamm, H. A. On the local Picard group, Tr. Mat. Inst. Steklova, Volume 267 (2009), pp. 138-145 (ISSN: 0371-9685) | DOI | MR | Zbl

Hochster, M.; Huneke, C. Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math., Volume 135 (1992), pp. 53-89 (ISSN: 0003-486X) | DOI | MR | Zbl

Huneke, C.; Lyubeznik, G. Absolute integral closure in positive characteristic, Adv. Math., Volume 210 (2007), pp. 498-504 (ISSN: 0001-8708) | DOI | MR | Zbl

Hamm, H. A.; Lê, D. T., Singularities in geometry and topology, World Sci. Publ., Hackensack, NJ, 2007, pp. 640-660 | DOI | MR | Zbl

Hochster, M. Some applications of the Frobenius in characteristic 0, Bull. Amer. Math. Soc., Volume 84 (1978), pp. 886-912 (ISSN: 0002-9904) | DOI | MR | Zbl

Hamm, H. A.; Tráng, L. D. Local generalizations of Lefschetz-Zariski theorems, J. reine angew. Math., Volume 389 (1988), pp. 157-189 (ISSN: 0075-4102) | DOI | MR | Zbl

Kleiman, S. L., Fundamental algebraic geometry (Math. Surveys Monogr.), Volume 123, Amer. Math. Soc., 2005, pp. 235-321 | MR

Kleiman, S. L. Toward a numerical theory of ampleness, Ann. of Math., Volume 84 (1966), pp. 293-344 (ISSN: 0003-486X) | DOI | MR | Zbl

Kollár, J. Grothendieck-Lefschetz type theorems for the local Picard group (preprint arXiv:1211.0317 ) | MR | Zbl

Langer, A. Semistable sheaves in positive characteristic, Ann. of Math., Volume 159 (2004), pp. 251-276 (ISSN: 0003-486X) | DOI | MR | Zbl

Roos, J.-E. Derived functors of inverse limits revisited, J. London Math. Soc., Volume 73 (2006), pp. 65-83 (ISSN: 0024-6107) | DOI | MR | Zbl

Ravindra, G. V.; Srinivas, V. The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic Geom., Volume 15 (2006), pp. 563-590 (ISSN: 1056-3911) | DOI | MR | Zbl

Siu, Y.-T. Extending coherent analytic sheaves, Ann. of Math., Volume 90 (1969), pp. 108-143 (ISSN: 0003-486X) | DOI | MR | Zbl

Swan, R. G., Algebra and geometry (Taipei, 1995) (Lect. Algebra Geom.), Volume 2, Int. Press, Cambridge, MA, 1998, pp. 135-192 | MR | Zbl

Tate, J., Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, 1997, pp. 121-154 | DOI | MR | Zbl

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