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Ample vector bundles on Fano manifolds. (English) Zbl 0744.14009
In response to some problems posed by S. Mukai [in Birational geometry of algebraic varieties - open problems. Report on a conference in Katata (1988)] the following theorems are proven:
(1): If \(X\) is a projective manifold of dimension \(n\), admitting an ample vector bundle of rank \(n+1\) with \(c_ 1(E)=c_ 1(X)\), then \(X=\mathbb{P}_ n\) and \(E\simeq{\mathcal O}_{\mathbb{P}_ n}(1)^{\oplus(n+1)}\).
(2): If \(X\) is a projective manifold of dimension \(n\) and \(E\) an ample vector bundle of rank \(n\) with \(c_ 1(E)=c_ 1(X)\). Then either \((X,E)\simeq(\mathbb{P}_ n,{\mathcal O}(2)\oplus{\mathcal O}(1)^{\oplus(n-1)})\) or \(\simeq(\mathbb{P}_ n,T_{\mathbb{P}_ n})\) or \(\simeq(Q_ n,{\mathcal O}_{Q_ n}(1)^{\oplus n})\), where \(Q_ n\) denotes the \(n\)-dimensional smooth quadric.
A consequence is a different proof of Mori’s theorem: Any projective manifold with ample tangent bundle is the projective space. — The idea of the proof is to analyze the extremal rays of \(P(E)\) and the associated contractions.
Reviewer: H.Lange (Erlangen)

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
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