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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
Article

Abstract

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 

Notes

Acknowledgments

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.

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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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