A characterization of \({\mathbb{P}}_ n\) by vector bundles (Q802686)

The following result [conjectured by \textit{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 \(n=3\).