Skip to main content
Log in

A characterization of symplectic Grassmannians

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

A Correction to this article was published on 07 February 2019

This article has been updated

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Change history

  • 07 February 2019

    We refer to our original paper, using the same notation.

  • 07 February 2019

    We refer to our original paper, using the same notation.

References

  1. Araujo, C.: Rational curves of minimal degree and characterizations of projective spaces. Math. Ann. 335(4), 937–951 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, C., Castravet, A.-M.: Polarized minimal families of rational curves and higher Fano manifolds. Am. J. Math. 134(1), 87–107 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Bonavero, L., Casagrande, C., Druel, S.: On covering and quasi-unsplit families of curves. J. Eur. Math. Soc. (JEMS) 9(1), 45–57 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  6. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)

    Book  MATH  Google Scholar 

  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)

  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)

  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)

  10. Hwang, J.-M.: On the degrees of Fano four-folds of Picard number 1. J. Reine Angew. Math. 556, 225–235 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

  12. Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8(1), 51–63 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kebekus, S.: Families of singular rational curves. J. Algebraic Geom. 11(2), 245–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kebekus, S., Kovács, S.J.: Are rational curves determined by tangent vectors? Ann. Inst. Fourier (Grenoble) 54(1), 53–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. LeBrun, C.: Fano manifolds, contact structures, and quaternionic geometry. Internat. J. Math. 6(3), 419–437 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  18. Mok, N.: Geometric structures and substructures on uniruled projective manifolds. In: Foliation Theory in Algebraic Geometry, Simons Symposia, pp. 103–148. Springer (2016)

  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)

  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. Occhetta, G., Solá Conde, L.E., Wiśniewski, J.A.: Flag bundles on Fano manifolds. J. Math. Pures Appl. (9) 106(4), 651–669 (2016)

  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)

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gianluca Occhetta.

Additional information

Gianluca Occhetta supported by PRIN project “Geometria delle varietà algebriche” Grant Number 2010S47ARA_010. Gianluca Occhetta and Luis E. Solá Conde supported by the Department of Mathematics of the University of Trento. Kiwamu Watanabe partially supported by JSPS KAKENHI Grant Number 26800002.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Occhetta, G., Solá Conde, L.E. & Watanabe, K. A characterization of symplectic Grassmannians. Math. Z. 286, 1421–1433 (2017). https://doi.org/10.1007/s00209-016-1807-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1807-6

Mathematics Subject Classification

Navigation