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Remark on the existence of stable fibre bundles on surfaces. (À propos de l’existence de fibrés stables sur les surfaces.) (French) Zbl 0848.14008

Let \(X\) be an arbitrary smooth projective surface over the complex numbers \(\mathbb{C}\). The authors show how to construct a stable vector bundle with arbitrary rank \(r\), arbitrary first Chern class \(c_1\) (viewed as an element of the Néron-Severi group \(NS (X))\), and sufficiently large second Chern class \(l\). Their proof is algebraic, showing that a generic deformation of a sufficiently singular “elementary diminution” of a fixed vector bundle \({\mathcal E}\) (with first Chern class \(c_1)\) is stable (and locally free if \(l\) is sufficiently large). This paper extends previous work by Y. Laszlo on sheaves on \(\mathbb{P}^2\) [Math. Ann. 299, No. 4, 597-608 (1994; Zbl 0846.14011)].
Several other authors have obtained similar results, including D. Gieseker and J. Li, Z. Qin and W. P. Li, and M. Maruyama.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0846.14011
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