A note on vector bundles on Hirzebruch surfaces
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.
scientific article; zbMATH DE number 5919183