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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
##### Keywords:
log-terminal singularities
Full Text:
##### References:
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