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Sur l’espace de modules des fibres de Yang et Mills. (French) Zbl 0532.14003
Mathématique et physique, Sémin. Éc. Norm. Supér., Paris 1979-1982, Prog. Math. 37, 65-137 (1983).
[For the entire collection see Zbl 0516.00021.]
A stable vector bundle E on \({\mathbb{P}}^ 3_{{\mathbb{C}}}\) with \(c_ 1(E)=0\) and \(H^ 1({\mathbb{P}}^ 3,E(-2))=0\) is called a mathematical instanton bundle, or ”fibré de Yang et Mills” in this paper. Denote by M(n) the moduli space for such bundles with rank \(=2\) and \(c_ 2=n\geq 1\). - The main result of the paper is that M(n) is smooth of dimension 8n-3 for \(n\leq 4\), the essential part of the proof being the estimate \(\dim Ext^ 1(E,E(-2))\leq \frac{1}{2}n(n-1)\). [W. Barth has shown that M(n) is irreducible for \(n\leq 4\), in Math. Ann. 258, 81-106 (1981; Zbl 0477.14014).] - The last part of the paper is concerned with the rational map \(res_ X:M(n)\to M_ X(2n)\) obtained by restricting bundles on \({\mathbb{P}}^ 3\) to a smooth quadric surface X. It is shown that \(res_ X\) is etale for \(n=1\) but not for \(n=2\). [G. Ellingsrud has shown that \(res_ X\) is not étale for any \(n\geq 2\), in ”Le fibré d’endomorphismes d’un instanton n’est pas necessairement un instanton”, Publ. de l’Institut de Recherche Mathématique Avancée (Juin 1981).]
Reviewer: S.A.Strømme

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces