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Generalized Mukai conjecture for special Fano varieties. (English) Zbl 1068.14049
A Fano variety \(X\) is a projective variety with ample anti-canonical bundle \(-K_X\). For such varieties, the Picard number \(\rho\) is strictly related to the geometry of rational curves on \(X\). Namely, defining the pseudo-index as \(i:=\min\{m: -K_X\cdot C=m\) for some rational curve \(C\subset X\}\), it has been conjectured that \( \rho\leq {\dim X}/(i-1)\) and equality implies that \(X\) is a product of projective spaces. The conjecture has been proved by J. A. Wisniewski [Manuscr. Math. 68, 135-141 (1990; Zbl 0715.14033)], when \(i>(\dim X+2)/2\), and by L. Bonavero, C. Casagrande, O. Debarre and S. Druel [Comment. Math. Helv. 78, No.3, 601–626 (2003; Zbl 1044.14019)], for varieties of dimension \(4\).
In the paper under review, the authors prove that the conjecture holds for \(i\geq (\dim X+3)/3\), provided that \(X\) admits an unsplit covering family of rational curves, i.e. a proper component of the space of rational curves Hom\(^n_{\text{bir}}(\mathbb P^1,X)\) mod Aut(\(\mathbb P^1\)), which dominates \(X\).
Using a careful analysis of split and unsplit families of rational curves on \(X\), the authors are then able to prove the conjecture for Fano varieties of dimension \(5\).

MSC:
14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
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[1] M. Andreatta and J.A. Wiśniewski: “On manifolds whose tangent bundle contains an ample subbundle”, Invent. Math., Vol. 146, (2001), pp. 209-217. http://dx.doi.org/10.1007/PL00005808 · Zbl 1081.14060
[2] L. Bonavero, C. Casagrande, O. Debarre and S. Druel: “Sur une conjecture de Mukai”, Comment. Math. Helv., Vol. 78, (2003), pp. 601-626. http://dx.doi.org/10.1007/s00014-003-0765-x · Zbl 1044.14019
[3] L. Bonavero, F. Campana and J.A. Wiśniewski: “Variétés complexes dont l’éclat’ee en un point est de Fano”, C.R. Math. Acad. Sci. Paris, Vol. 334, (2002), pp. 463-468. · Zbl 1036.14020
[4] F. Campana: “Connexité rationnelle des variétés de Fano”, Ann. Sci. École Norm. Sup., Vol. 25, (1992), pp. 539-545. · Zbl 0783.14022
[5] K. Cho, Y. Miyaoka and N.I. Shepherd-Barron: “Characterizations of projective space and applications to complex symplectic manifolds”, in: Higher dimensional birational geometry (Kyoto, 1997) Adv. Stud. Pure Math., Vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1-88. · Zbl 1063.14065
[6] O. Debarre: Higher-Dimensional Algebraic Geometry, Universitext Springer-Verlag, New York, 2001. · Zbl 0978.14001
[7] S. Kebekus: “Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron”, In: Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 147-155. · Zbl 1046.14028
[8] J. Kollár: Rational Curves on Algebraic Varieties, Ergebnisse der Math. Vol. 32, Springer-Verlag, 1996.
[9] J. Kollár, Y. Miyaoka and S. Mori: “Rational connectedness and boundedness of Fano manifolds”, J. Diff. Geom. Vol. 36, (1992), pp. 765-779. · Zbl 0759.14032
[10] S. Mori: “Projective manifolds with ample tangent bundle”, Ann. Math., Vol. 110, (1979), pp. 595-606. http://dx.doi.org/10.2307/1971241 · Zbl 0423.14006
[11] S. Mukai: “Open problems”, In: Birational geometry of algebraic varieties, Taniguchi Foundation, Katata, 1988.
[12] G. Occhetta: A characterization of products of projective spaces, preprint, February 2003, http://www.science.unitn.it/∼occhetta.
[13] J.A. Wiśniewski: “On a conjecture of Mukai”, Manuscripta Math., Vol. 68, (1990), pp. 135-141.
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