Comptes Rendus
no. 6
Fiberwise stable bundles on elliptic threefolds with relative Picard number one
[Fibrés vectoriels relativement stables sur variétés elliptiques de dimension trois dont le nombre relatif de Picard est un]
Andrei Căldăraru1
1 Mathematics Department, University of Massachusetts, Amherst, MA 01003, USA
Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 469-472.

Nous prouvons que les fibrés vectoriels relativement stables sont préservés par des transformées de Fourier–Mukai entre variétés elliptiques de dimension trois dont le nombre relatif de Picard est un. En utilisant ces fibrés nous définissons des nouveaux invariants de variétés elliptiques, et nous étudions la relation entre les invariants d'une variété et ceux d'un éspace relatif de modules des fibrés stables sur elle. Ces résultats nous permettent de calculer la forme d'intersection sur un certain nouvel exemple de variété de Calabi–Yau de dimension trois.

We show that fiberwise stable vector bundles are preserved by relative Fourier–Mukai transforms between elliptic threefolds with relative Picard number one. Using these bundles we define new invariants of elliptic fibrations, and we relate the invariants of a space with those of a relative moduli space of stable sheaves on it. As a byproduct, we calculate the intersection form of a certain new example of an elliptic Calabi–Yau threefold.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(02)02290-2
Andrei Căldăraru 1

1 Mathematics Department, University of Massachusetts, Amherst, MA 01003, USA 
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     pages = {469--472},
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Andrei Căldăraru. Fiberwise stable bundles on elliptic threefolds with relative Picard number one. Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 469-472. doi : 10.1016/S1631-073X(02)02290-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02290-2/

[1] M.F. Atiyah Vector bundles over an elliptic curve, Proc. London Math. Soc., Volume 7 (1957) no. 3, pp. 414-452

[2] C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, J. Muñoz Porras, Relatively stable bundles over elliptic fibrations, Preprint, alg-geom 0109123

[3] T. Bridgeland Fourier–Mukai transforms for elliptic surfaces, J. reine angew. Math., Volume 498 (1998), pp. 115-133

[4] T. Bridgeland, A. Maciocia, Fourier–Mukai transforms for K3 fibrations, Preprint, alg-geom/9908022

[5] T. Bridgeland, Flops and derived categories, Preprint, alg-geom/0009053

[6] A. Căldăraru, Derived categories of twisted sheaves on Calabi–Yau manifolds, Ph.D. thesis, Cornell University, 2000

[7] A. Căldăraru, Derived categories of twisted sheaves on elliptic threefolds, J. reine angew. Math. (to appear)

[8] L.W. Tu Semistable bundles over an elliptic curve, Adv. Math., Volume 98 (1993) no. 1, pp. 1-26

[9] C.T.C. Wall Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math., Volume 1 (1966), pp. 355-374 Corrigendum, ibid 2 (306) (1966)

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