A Note on vector bundles on Hirzebruch surfaces

Marian Aprodu; Marius Marchitan

doi:10.1016/j.crma.2011.04.013

Algebraic Geometry

A Note on vector bundles on Hirzebruch surfaces
[Une Note sur des fibrés vectoriels sur des surfaces de Hirzebruch]

Présenté par : Claire Voisin

Marian Aprodu ¹ ; Marius Marchitan ²

¹ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
² University “Ştefan cel Mare”, Str. Universităţii 13, 720229 Suceava, Romania

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 687-690.

Résumé
Abstract

Dans la littérature, deux méthodes de construction fondamentales ont été utilisées pour étudier les fibrés vectoriels sur une surface de Hirzebruch. Dʼune part, nous avons la méthode de Serre et les modifications élémentaires, décrivant dʼune manière canonique les fibrés de rang deux comme des extensions (Brînzănescu et Stoia, 1984 [4,5], Brînzănescu, 1996 [6], Brosius, 1983 [7], Friedman, 1998 [9]) et dʼautre part, nous avons la suite spectrale de Beilinson (Buchdahl, 1987 [8]). Moralement, la suite spectrale de Beilinson nous indique comment récupérer un fibré à partir de la cohomologie de ses tensorisations et de certains morphismes de faisceaux (les différentielles de la suite spectrale). Le but de cette Note est de montrer que lʼextension canonique dʼun fibré de rang deux peut être déduite de la suite spectrale de Beilinson dʼune tensorisation convenable, appellée la normalisation. Dans la dernière partie, nous donnons un critère cohomologique pour quʼun fibré vectoriel topologiquement trivial sur une surface de Hirzebruch soit trivial. Afin de souligner les relations et les différences entre les deux méthodes de construction mentionnées ci-dessus, deux démonstrations différentes sont présentées.

In literature, two basic construction methods have been used to study vector bundles on a Hirzebruch surface. On the one hand, we have Serreʼs method and elementary modifications, describing rank-2 bundles as extensions in a canonical way (Brînzănescu and Stoia, 1984 [4,5], Brînzănescu, 1996 [6], Brosius, 1983 [7], Friedman, 1998 [9]), and on the other hand, we have a Beilinson-type spectral sequence (Buchdahl, 1987 [8]). Morally, the Beilinson spectral sequence indicates how to recover a bundle from the cohomology of its twists and from some sheaf morphisms (the differentials of the sequence). The aim of this Note is to show that the canonical extension of a rank-2 bundle can be deduced from the Beilinson spectral sequence of a suitable twist, called the normalization. In the final part we give a cohomological criterion for a topologically trivial vector bundle on a Hirzebruch surface to be trivial. To emphasize the relations and the differences between these two construction methods mentioned above, two different proofs are given.

Reçu le : 2011-03-24
Accepté le : 2011-04-26
Publié le : 2011-05-23

DOI : 10.1016/j.crma.2011.04.013

Affiliations des auteurs :

Marian Aprodu ¹ ; Marius Marchitan ²

¹ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
² University “Ştefan cel Mare”, Str. Universităţii 13, 720229 Suceava, Romania

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     title = {A {Note} on vector bundles on {Hirzebruch} surfaces},
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Marian Aprodu; Marius Marchitan. A Note on vector bundles on Hirzebruch surfaces. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 687-690. doi : 10.1016/j.crma.2011.04.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.013/

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