Rational connectivity of Fano varieties. (Connexité rationnelle des variétés de Fano.) (French) Zbl 0783.14022

It is shown that a Fano variety \(X\), i.e., a nonsingular projective variety with \(-K_ X\) ample, is rationally connected. Namely, for any two points \(x,y\) of \(X\) there is a connected chain of rational curves \(C=\bigcup_{1 \leq i \leq n}C_ i\) such that \(x\), \(y \in C\). This result was also proved by J. Kollar, Y. Miyaoka and S. Mori [J. Differ. Geom. 36, No. 3, 765-779 (1992; Zbl 0759.14032)].


14J45 Fano varieties
14M20 Rational and unirational varieties


Zbl 0759.14032
Full Text: DOI Numdam EuDML


[1] F. CAMPANA , Coréduction algébrique d’un espace analytique compact faiblement Kählerien (Inv. Math., vol. 81, 1981 , p. 187-223). MR 84e:32028 | Zbl 0436.32024 · Zbl 0436.32024 · doi:10.1007/BF01393876
[2] F. CAMPANA , Théorème de finitude pour les variétés de Fano suffisamment uniréglées [J.f.d.R.u.A. Math. (à paraître)]. · Zbl 0931.14023
[3] J. KOLLÁR , Y. MIYAOKA et S. MORI , Rational Curves on Fano Manifolds , Preprint. · Zbl 0776.14012
[4] J. KOLLÁR , Y. MIYAOKA et S. MORI , Rationally Connected Varieties , Preprint. · Zbl 0780.14026
[5] Y. MIYAOKA , On the Structure of Uniruled Varieties , Manuscrit non publié 1986 .
[6] Y. MIYAOKA et S. MORI , A numerical Criterion for Uniruledness (Ann. Math., vol. 124, 1986 , p. 65-69). MR 87k:14046 | Zbl 0606.14030 · Zbl 0606.14030 · doi:10.2307/1971387
[7] S. MORI , Projective Manifolds with Ample Tangent Bundles (Ann. Math., vol. 110, 1979 , p. 593-606). MR 81j:14010 | Zbl 0423.14006 · Zbl 0423.14006 · doi:10.2307/1971241
[8] A. NADEL , Boundedness of Fano Varieties with Picard Number One , Preprint. · Zbl 0754.14026 · doi:10.2307/2939285
[9] H. TSUJI , Boundedness of the Degree of Fano Manifolds with b2 = 1 , Preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.