A characterization of \({\mathbb{P}}_ n\) by vector bundles. (English) Zbl 0726.14034

Peternell, Thomas

Math. Z. 205, No. 3, 487-490 (1990).

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 \(n=3\).

Reviewer: O.Păsărescu (Bucureşti)

MSC:

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

Keywords:

characterization of projective space; first Chern class; extremal rational curves; ample vector bundle

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