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Ample vector bundles on singular varieties. (English) Zbl 0842.14010
Let \(X\) be an \(n\)-dimensional projective variety having at most log-terminal singularities and let \(E\) be an ample vector bundle of rank \(r\) on \(X\). The author proves that:
(1) If \(r= n+1\) and \(c_1 (X)= c_1 (E)\) then \((X, E) \simeq (\mathbb{P}^n, {\mathcal O}_P (1)^{n+ 1})\); and
(2) If \(r\geq n+1\) then \(K_X+ c_1 (E)\) is ample unless \((X, E)\simeq (\mathbb{P}^n, {\mathcal O}_P (1)^{n+ 1})\).
If \(X\) is smooth, the results where already known [cf. Y.-G. Ye and Q. Zhang, Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011) and T. Peternell [Math. Z. 205, No. 3, 487-490 (1990; Zbl 0726.14034)]. However, the argument used in the smooth case do not work in the singular one.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
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References:
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