A note on vector bundles on Hirzebruch surfaces
type: Item
instance of: scholarly article
title: A note on vector bundles on Hirzebruch surfaces
zbMATH Open document ID: 1243.14033
author: Marian Aprodu
author: Marius Marchitan
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2011-07-08T00:00:00Z
review text: Let \(H\) be a Hirzebruch surface. Earlier results had shown constructions of rank-two vector bundles on \(H\) as extensions of two line bundles [\textit{V. Brînzănescu} and \textit{M. Stoia}, Rev. Roum. Math. Pures Appl. 29, 661--673 (1984; Zbl 0547.14005); \textit{V. Brînzănescu} and \textit{M. Stoia}, Lect. Notes Math. 1056, 34--46 (1984; Zbl 0547.14006); \textit{V. Brînzănescu}, Holomorphic vector bundles over compact complex surfaces. Lecture Notes in Mathematics. 1624. Berlin: Springer-Verlag (1996; Zbl 0848.32024); \textit{J.~E. Brosius}, Math. Ann. 265, 155--168 (1983; Zbl 0503.55012); \textit{R. Friedman}, Algebraic surfaces and holomorphic vector bundles. Universitext. New York, NY: Springer (1998; Zbl 0902.14029)], and by means of a spectral sequence [\textit{N.~P. Buchdahl}, Math. Z. 194, 143--152 (1987; Zbl 0627.14028)]. The authors compare these two ways of construction and their main result is that the first case is a particular case of the latter, if for a given rank-two vector bundle \(M\) we consider its normalisation. They conclude by showing a cohomological criterion for the tiviality of a topologically trivial vector bundle on \(H\) (i.e. a vector bundle whose Chern classes vanish). They present two proofs, one for arbitrary rank, using the Beilinson spectral sequence, and another for rank two, using extensions of line bundles.
reviewed by: Pedro Macias Marques
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J26
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18G40
zbMATH DE Number: 5919183
zbMATH Keywords: vector bundles
zbMATH Keywords: Hirzebruch surface
zbMATH Keywords: Beilinson spectral sequence
MaRDI profile type: Publication
full work available at URL: j.crma.2011.04.013
OpenAlex ID: W2020354948
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DOI: 10.1016/J.CRMA.2011.04.013
description: scientific article; zbMATH DE number 5919183