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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)].

MSC:
14J45 Fano varieties
14M20 Rational and unirational varieties
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References:
[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
[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
[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
[8] A. NADEL , Boundedness of Fano Varieties with Picard Number One , Preprint. · Zbl 0754.14026
[9] H. TSUJI , Boundedness of the Degree of Fano Manifolds with b2 = 1 , Preprint.
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