Mathematische Zeitschrift

, Volume 286, Issue 3–4, pp 1421–1433 | Cite as

A characterization of symplectic Grassmannians

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


We provide a characterization of symplectic Grassmannians in terms of their varieties of minimal rational tangents.

Mathematics Subject Classification

Primary 14J45 Secondary 14E30 14M15 14M17 



The authors would like to thank J.A. Wiśniewski for the interesting discussions they had on this topic, during his visit to the University of Trento in 2016.


  1. 1.
    Araujo, C.: Rational curves of minimal degree and characterizations of projective spaces. Math. Ann. 335(4), 937–951 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Araujo, C., Castravet, A.-M.: Polarized minimal families of rational curves and higher Fano manifolds. Am. J. Math. 134(1), 87–107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1984)Google Scholar
  4. 4.
    Bonavero, L., Casagrande, C., Druel, S.: On covering and quasi-unsplit families of curves. J. Eur. Math. Soc. (JEMS) 9(1), 45–57 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cho, K., Miyaoka, Y., Shepherd-Barron, N.I.: Characterizations of projective space and applications to complex symplectic manifolds. In: Higher Dimensional Birational Geometry (Kyoto, 1997), volume 35 of Adv. Stud. Pure Math., pp. 1–88. Math. Soc. Japan, Tokyo (2002)Google Scholar
  6. 6.
    Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fujita, T.: On polarized manifolds whose adjoint bundles are not semipositive. In: Algebraic Geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pp. 167–178. North-Holland, Amsterdam (1987)Google Scholar
  8. 8.
    Hong, J., Hwang, J.-M.: Characterization of the rational homogeneous space associated to a long simple root by its variety of minimal rational tangents. In: Algebraic Geometry in East Asia—Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pp. 217–236. Math. Soc. Japan, Tokyo (2008)Google Scholar
  9. 9.
    Hwang, J.-M.: Geometry of Minimal Rational Curves on Fano Manifolds. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pp. 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2001)Google Scholar
  10. 10.
    Hwang, J.-M.: On the degrees of Fano four-folds of Picard number 1. J. Reine Angew. Math. 556, 225–235 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hwang, J.-M.: Mori geometry meets Cartan geometry: varieties of minimal rational tangents. In: Proceedings of the International Conference of Mathematicians, Seoul, 2014, vol. I, pp. 369–394. Kyung Moon SA Co., Ltd., Seoul (2014)Google Scholar
  12. 12.
    Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8(1), 51–63 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kebekus, S.: Families of singular rational curves. J. Algebraic Geom. 11(2), 245–256 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kebekus, S., Kovács, S.J.: Are rational curves determined by tangent vectors? Ann. Inst. Fourier (Grenoble) 54(1), 53–79 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kollár, J.: Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (1996)Google Scholar
  16. 16.
    LeBrun, C.: Fano manifolds, contact structures, and quaternionic geometry. Internat. J. Math. 6(3), 419–437 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mok, N.: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. In: Third International Congress of Chinese Mathematicians. Part 1, 2, volume 2 of AMS/IP Stud. Adv. Math., 42, pt. 1, pp. 41–61. Amer. Math. Soc., Providence, RI (2008)Google Scholar
  18. 18.
    Mok, N.: Geometric structures and substructures on uniruled projective manifolds. In: Foliation Theory in Algebraic Geometry, Simons Symposia, pp. 103–148. Springer (2016)Google Scholar
  19. 19.
    Muñoz, R., Occhetta, G., Solá Conde, L.E., Watanabe, K., Wiśniewski, J.A.: A survey on the Campana–Peternell conjecture. Rend. Istit. Mat. Univ. Trieste 47, 127–185 (2015)Google Scholar
  20. 20.
    Occhetta, G., Solá Conde, L.E., Watanabe, K., Wiśniewski, J.A.: Fano manifolds whose elementary contractions are smooth \(\mathbb{P}^1\)-fibrations. Ann. Sci. Norm. Super. Pisa, cl. Sci. (5), 2017. Preprint arXiv:1407.3658
  21. 21.
    Occhetta, G., Solá Conde, L.E., Wiśniewski, J.A.: Flag bundles on Fano manifolds. J. Math. Pures Appl. (9) 106(4), 651–669 (2016)Google Scholar
  22. 22.
    Siu, Y.T.: Errata: Nondeformability of the complex projective space [J. Reine Angew. Math. 399: 208–219; MR1004139 (90h:32048)]. J. Reine Angew. Math. 431(65–74), 1992 (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoPovo di TrentoItaly
  2. 2.Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and EngineeringSaitama UniversitySakura-kuJapan

Personalised recommendations