an:00799547
Zbl 0842.14010
Zhang, Qi
Ample vector bundles on singular varieties
EN
Math. Z. 220, No. 1, 59-64 (1995).
00028348
1995
j
14F05 14E15 14B05
log-terminal singularities
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. \textit{Y.-G. Ye} and \textit{Q. Zhang}, Duke Math. J. 60, No. 3, 671-687 (1990; Zbl 0709.14011) and \textit{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.
I.Coand?? (Bucure??ti)
Zbl 0709.14011; Zbl 0726.14034