Mathematische Annalen

, Volume 361, Issue 3–4, pp 583–609 | Cite as

Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle

  • Roberto Muñoz
  • Gianluca OcchettaEmail author
  • Luis E. Solá Conde
  • Kiwamu Watanabe


A Fano manifold \(X\) with nef tangent bundle is of Flag-Type if it has the same kind of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram \(\mathcal {D}(X)\) with any such \(X\), based on the numerical properties of its contractions. We then show that \(\mathcal {D}(X)\) is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana–Peternell conjecture holds when \(X\) is a Flag-Type manifold whose Dynkin diagram is \(A_n\), i.e. we show that \(X\) is the variety of complete flags of linear subspaces in \(\mathbb {P}^n\).

Mathematics Subject Classification

Primary 14M15 Secondary 14E30 14J45 



This project was conceived when the third and fourth author enjoyed a grant of the “Research in pairs” program of the Fondazione Bruno Kessler, at the Centro Internazionale per la Ricerca Matematica (Trento). We would like to express our gratitude to this institution for its support and hospitality. We would also like to thank J. Wiśniewski for his interest in our project and his useful comments.


  1. 1.
    Chierici, E., Occhetta, G.: The cone of curves of Fano varieties of coindex four. Int. J. Math. 17, 1195–1221 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Campana, F., Peternell, T.: Projective manifolds whose tangent bundles are numerically effective. Math. Ann. 289(1), 169–187 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Campana, F., Peternell, T.: \(4\) -folds with numerically effective tangent bundles and second Betti numbers greater than one. Manuscr. Math. 79(3–4), 225–238 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Casagrande, C.: Quasi-elementary contractions of Fano manifolds. Compos. Math. 144, 1429–1460 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Debarre, O.: Higher-dimensional algebraic geometry. Universitext. Springer, New York (2001)CrossRefGoogle Scholar
  6. 6.
    Demailly, J.P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3(2), 295–345 (1994)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)Google Scholar
  9. 9.
    Hwang, J.-M.: Rigidity of rational homogeneous spaces. In: International Congress of Mathematicians, vol. II, pp. 613–626. European Mathematical Society, Zürich (2006)Google Scholar
  10. 10.
    Kac, V.G.: Infinite dimensional Lie algebras: an introduction. Progress in Mathematics, vol. 44. Birkhauser, Boston (1983)Google Scholar
  11. 11.
    Kempf, G.R.: Linear systems on homogeneous spaces. Ann. Math. (2) 103(3), 557–591 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(1), 65–100 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lau, C.-H.: Holomorphic maps from rational homogeneous spaces onto projective manifolds. J. Algebraic Geom. 18(2), 223–256 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Mok, N.: On Fano manifolds with nef tangent bundles admitting \(1\)-dimensional varieties of minimal rational tangents. Trans. Am. Math. Soc. 354(7), 2639–2658 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110(3), 593–606 (1979)CrossRefzbMATHGoogle Scholar
  16. 16.
    Muñoz, R., Occhetta, G., Solá Conde, L.E.: On rank \(2\) vector bundles on Fano manifolds. Kyoto J. Math. 54(1), 167–197 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Occhetta, G., Solá Conde, L.E., Watanabe, K., Wisniewski, J.A.: Fano manifolds whose elementary contractions are smooth \({\mathbb{P}}^1\)-fibrations. arXiv:1407.3658
  18. 18.
    Occhetta, G., Wiśniewski, J.A.: On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties. Math. Z. 241(1), 35–44 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Solá Conde, L.E., Wiśniewski, J.A.: On manifolds whose tangent bundle is big and \(1\)-ample. Proc. Lond. Math. Soc. (3) 89(2), 273–290 (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    Tsaranov, S.V.: Representation and classification of Coxeter monoids. Eur. J. Comb. 11(2), 189–204 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Yasutake, K.: On projective space bundles with nef normalized tautological divisor. arXiv:1104.5084v2
  22. 22.
    Watanabe, K.: \({\mathbb{P}}^1\)-bundles admitting another smooth morphism of relative dimension one. J. Algebra 414(15), 105–119 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Watanabe, K.: Fano \(5\)-folds with nef tangent bundles and Picard numbers greater than one. Math. Z. 276(1–2), 39–49 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Roberto Muñoz
    • 1
  • Gianluca Occhetta
    • 2
    Email author
  • Luis E. Solá Conde
    • 3
  • Kiwamu Watanabe
    • 4
  1. 1.Departamento de Matemática Aplicada, ESCETUniversidad Rey Juan CarlosMóstolesSpain
  2. 2.Dipartimento di MatematicaUniversità di TrentoPovo, Trento (TN)Italy
  3. 3.Korea Institute for Advanced StudySeoulKorea
  4. 4.Graduate School of Science and EngineeringSaitama UniversitySaitamaJapan

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