European Journal of Mathematics

, Volume 5, Issue 3, pp 858–871 | Cite as

The generalized Mukai conjecture for toric log Fano pairs

  • Kento FujitaEmail author
Research Article


We prove the generalized Mukai conjecture for (not necessarily \({\mathbb {Q}}\)-factorial) toric log Fano pairs.


Fano varieties Toric varieties Mukai conjecture 

Mathematics Subject Classification

14J45 14M25 



The author thanks the referee for helpful comments.


  1. 1.
    Andreatta, M., Chierici, E., Occhetta, G.: Generalized Mukai conjecture for special Fano varieties. Cent. Eur. J. Math. 2(2), 272–293 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3(3), 493–535 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonavero, L., Casagrande, C., Debarre, O., Druel, S.: Sur une conjecture de Mukai. Comment. Math. Helv. 78(3), 601–626 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casagrande, C.: The number of vertices of a Fano polytope. Ann. Inst. Fourier (Grenoble) 56(1), 121–130 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  6. 6.
    Eikelberg, M.: The Picard group of a compact toric variety. Results Math. 22(1–2), 509–527 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fujino, O.: Notes on toric varieties from Mori theoretic viewpoint. Tohoku Math. J. 55(4), 551–564 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fujino, O.: Non-vanishing theorem for log canonical pairs. J. Algebraic Geom. 20(4), 771–783 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fujita, K.: The Mukai conjecture for log Fano manifolds. Cent. Eur. J. Math. 12(1), 14–27 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gagliardi, G., Hofscheier, J.: The generalized Mukai conjecture for symmetric varieties. Trans. Amer. Math. Soc. 369(4), 2615–2649 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. With the Collaboration of C.H. Clemens and A. Corti. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. 110(3), 593–606 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mukai, S.: Problems on characterization of the complex projective space. In: Birational Geometry of Algebraic Varieties. Open Problems. Proceedings of the 23rd Symposium of the Taniguchi Foundation, pp. 57–60. Taniguchi Foundation, Katata (1988)Google Scholar
  15. 15.
    Novelli, C., Occhetta, G.: Rational curves and bounds on the Picard number of Fano manifolds. Geom. Dedicata 147, 207–217 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

Personalised recommendations