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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
##### Keywords:
Fano manifolds; ample vector bundle; extremal rays
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