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A characterization of $${\mathbb{P}}_ n$$ by vector bundles. (English) Zbl 0726.14034
The following result [conjectured by S. Mukai; cf. “Open problems. Classification of algebraic and analytic manifolds”, Proc. Symp., Katata/Jap. 1982, Prog. Math. 39, 591-630 (1983; Zbl 0527.14002)] is proved:
Theorem: Let X be a compact complex manifold of dimension n, E an ample vector bundle on X of rank $$(n+1)$$ satisfying $$c_ 1(E)=c_ 1(X)$$. Then $$X\cong P_ n$$ and $$E\cong {\mathcal O}_{P_ n}(1)^{n+1}.$$
The cases $$n\leq 2$$ are clear. Mukai proved the case $$n=3$$.

##### MSC:
 14N05 Projective techniques in algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 57R20 Characteristic classes and numbers in differential topology
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##### References:
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