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**Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron.**
*(English)*
Zbl 1046.14028

Bauer, Ingrid (ed.) et al., Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer (ISBN 3-540-43259-0/hbk). 147-155 (2002).

From the introduction: The aim of this paper is to give a short proof of the following characterization of the projective space.

Theorem. Let \(X\) be a projective manifold of dimension \(n\geq 3\), defined over the field \(\mathbb{C}\) of complex numbers. Assume that for every curve \(C\subset X\), we have \(-K_X.C\geq n+1\). Then \(X\) is isomorphic to the projective space.

A proof was first given in a preprint by K. Cho, Y. Miyaoka and N. Shepherd-Barron [Adv. Stud. Pure Math. 35, 1–88 (2002; Zbl 1063.14065)]. While the proof in the present paper is shorter, involves substantial technical simplifications and is perhaps more transparent, the essential ideas are taken from that paper.

For the entire collection see [Zbl 0989.00069].

Theorem. Let \(X\) be a projective manifold of dimension \(n\geq 3\), defined over the field \(\mathbb{C}\) of complex numbers. Assume that for every curve \(C\subset X\), we have \(-K_X.C\geq n+1\). Then \(X\) is isomorphic to the projective space.

A proof was first given in a preprint by K. Cho, Y. Miyaoka and N. Shepherd-Barron [Adv. Stud. Pure Math. 35, 1–88 (2002; Zbl 1063.14065)]. While the proof in the present paper is shorter, involves substantial technical simplifications and is perhaps more transparent, the essential ideas are taken from that paper.

For the entire collection see [Zbl 0989.00069].

### MSC:

14N05 | Projective techniques in algebraic geometry |

14M20 | Rational and unirational varieties |

14E08 | Rationality questions in algebraic geometry |

14E30 | Minimal model program (Mori theory, extremal rays) |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |