an:04198193
Zbl 0726.14034
Peternell, Thomas
A characterization of \({\mathbb{P}}_ n\) by vector bundles
EN
Math. Z. 205, No. 3, 487-490 (1990).
00156065
1990
j
14N05 14F05 57R20
characterization of projective space; first Chern class; extremal rational curves; ample vector bundle
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 \(n=3\).
O.P??s??rescu (Bucure??ti)
Zbl 0527.14002