A note on vector bundles on Hirzebruch surfaces. (English) Zbl 1243.14033
Let \(H\) be a Hirzebruch surface. Earlier results had shown constructions of rank-two vector bundles on \(H\) as extensions of two line bundles [V. Brînzănescu and M. Stoia, Rev. Roum. Math. Pures Appl. 29, 661–673 (1984; Zbl 0547.14005); V. Brînzănescu and M. Stoia, Lect. Notes Math. 1056, 34–46 (1984; Zbl 0547.14006); V. Brînzănescu, Holomorphic vector bundles over compact complex surfaces. Lecture Notes in Mathematics. 1624. Berlin: Springer-Verlag (1996; Zbl 0848.32024); J. E. Brosius, Math. Ann. 265, 155–168 (1983; Zbl 0503.55012); R. Friedman, Algebraic surfaces and holomorphic vector bundles. Universitext. New York, NY: Springer (1998; Zbl 0902.14029)], and by means of a spectral sequence [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.
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.
Reviewer: Pedro Macias Marques (Évora)
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14J26 | Rational and ruled surfaces |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 18G40 | Spectral sequences, hypercohomology |
Citations:
Zbl 0547.14005; Zbl 0547.14006; Zbl 0848.32024; Zbl 0503.55012; Zbl 0902.14029; Zbl 0627.14028References:
| [1] | Aprodu, M.; Brînzănescu, V., Fibrés vectoriels de rang 2 sur les surfaces réglées, C. R. Acad. Sci. Paris, Ser. I, 323, 6, 627-630 (1996) · Zbl 0872.14036 |
| [2] | Aprodu, M.; Brînzănescu, V., Stable rank-2 vector bundles over ruled surfaces, C. R. Acad. Sci. Paris, Ser. I, 325, 3, 295-300 (1997) · Zbl 0905.14025 |
| [3] | Aprodu, M.; Brînzănescu, V., Beilinson type spectral sequences on scrolls, (Moduli Spaces and Vector Bundles. Moduli Spaces and Vector Bundles, London Math. Soc. Lecture Note Ser., vol. 359 (2009), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 426-436 · Zbl 1187.14051 |
| [4] | Brînzănescu, V.; Stoia, M., Topologically trivial algebraic 2-vector bundles on ruled surfaces. I, Rev. Roumaine Math. Pures Appl., 29, 8, 661-673 (1984) · Zbl 0547.14005 |
| [5] | Brînzănescu, V.; Stoia, M., Topologically trivial algebraic 2-vector bundles on ruled surfaces. II, (Algebraic Geometry. Algebraic Geometry, Bucharest, 1982. Algebraic Geometry. Algebraic Geometry, Bucharest, 1982, Lecture Notes in Math., vol. 1056 (1984), Springer: Springer Berlin), 34-46 · Zbl 0547.14006 |
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| [7] | Brosius, J. E., Rank-2 vector bundles on a ruled surface. I, Math. Ann., 265, 2, 155-168 (1983) · Zbl 0503.55012 |
| [8] | Buchdahl, N. P., Stable 2-bundles on Hirzebruch surfaces, Math. Z., 194, 1, 143-152 (1987) · Zbl 0627.14028 |
| [9] | Friedman, R., Algebraic Surfaces and Holomorphic Vector Bundles, Universitext (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0902.14029 |
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| [11] | Pragacz, P.; Srinivas, V.; Pati, V.; Yau, S. T.; etal., Diagonal subschemes and vector bundles, Special Volume Dedicated to J.-P. Serre on His 80th Birthday. Special Volume Dedicated to J.-P. Serre on His 80th Birthday, Pure Appl. Math. Quart., 4, 4, Part 1, 1233-1278 (2008) · Zbl 1157.14007 |
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