Skip to main content
SpringerLink
Log in

Contact Kähler Manifolds: Symmetries and Deformations

  • Conference paper
  • First Online:
Algebraic and Complex Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 71))

Abstract

We study complex compact Kähler manifolds X carrying a contact structure. If X is almost homogeneous and b 2(X) ≥ 2, then X is a projectivised tangent bundle (this was known in the projective case even without assumption on the existence of vector fields). We further show that a global projective deformation of the projectivised tangent bundle over a projective space is again of this type unless it is the projectivisation of a special unstable bundle over a projective space. Examples for these bundles are given in any dimension.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Andreatta, J. Wiśniewski, A view on contractions of higher-dimensional varieties, in Algebraic Geometry–Santa Cruz 1995. Proceedings of Symposia in Pure Mathematics, vol. 62, Part 1 (American Mathematical Society, Providence, 1997), pp. 153–183

    Google Scholar 

  2. C. Araujo, J. Ramón-Marí, Flat deformations of \(\mathbb{P}^{n}\). arXiv.AG/1212.3593v1

    Google Scholar 

  3. D. Barlet, Th. Peternell, M. Schneider, On two conjectures of Hartshorne’s. Math. Ann. 286, 13–25 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Barth, E. Oeljeklaus, Über die Albanese-Abbildung einer fast-homogenen Kählermannigfaltigkeit. Math. Ann. 211, 47–62 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Beauville, Fano contact manifolds and nilpotent orbits. Commun. Math. Helv. 73, 566–583 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Beltrametti, A.J. Sommese, The Adjunction Theory of Complex Projective Varieties (de Gruyter, Berlin, 1995)

    Book  MATH  Google Scholar 

  7. J. Bingener, On deformations of Kähler spaces II. Arch. d. Math. 41, 517–530 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Borel, R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten. Math. Ann. 145, 429–439 (1961/1962)

    Google Scholar 

  9. J.P. Demailly, On the Frobenius integrability of certain holomorphic p-forms, in Complex Geometry, Göttingen, 2000, ed. by I. Bauer et al. (Springer, Berlin, 2002), pp. 93–98

    Google Scholar 

  10. G. Elencwajg, The Brauer Groups in Complex Geometry. Lecture Notes in Mathematics, vol. 917 (Springer, Berlin/New York, 1982), pp. 222–230

    Google Scholar 

  11. T. Fujita, On polarized manifolds whose adjoint bundles are not semipositive. Adv. Stud. Pure Math. 10, 167–178 (1985)

    Google Scholar 

  12. T. Fujita, Classification Theories of Polarized Varieties. London Mathematical Society Lecture Notes Series, vol. 155 (Cambridge University Press, Cambridge/New York, 1990)

    Google Scholar 

  13. N. Goldstein, Ampleness in complex homogeneous spaces and a second Lefschetz theorem. Pac. J. Math. 106(2), 271–291 (1983)

    Article  MATH  Google Scholar 

  14. F. Hirzebruch, K. Kodaira, On the complex projective spaces. J. Math. Pures Appl. 36, 201–216 (1957)

    MathSciNet  MATH  Google Scholar 

  15. A. Höring, Uniruled varieties with split tangent bundle. Math. Z. 256(3), 465–479 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Husemoller, Fibre Bundles, 3rd edn. (Springer, New York, 1994)

    Book  Google Scholar 

  17. J.M. Hwang, Rigidity of rational homogeneous spaces, in International Congress of Mathematicians, Madrid, ed. by M. Sanz Solé, vol. II (European Mathematical Society, Zürich, 2006), pp. 613–626

    Google Scholar 

  18. J.M. Hwang, Deformations of the space of lines on the 5-dimensional hyperquadric (2010, Preprint)

    Google Scholar 

  19. S. Kebekus, Lines on contact manifolds. J. Reine Angew. Math. 539, 167–177 (2001)

    MathSciNet  MATH  Google Scholar 

  20. S. Kebekus, Th. Peternell, A.J. Sommese, J. Wisniewski, Projective contact manifolds. Invent. Math. 142, 1–15 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Kollár, S. Mori, Classification of three-dimensional flips. J. Am. Math. Soc. 5(3), 533–703 (1992)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Mukai, H. Umemura, Minimal rational threefolds, in Algebraic Geometry, Tokyo/Kyoto, 1982. Lecture Notes in Mathematics, vol. 1016 (Springer, Berlin, 1983), pp. 490–518

    Google Scholar 

  24. N. Nakayama, The lower semicontinuity of the plurigenera of complex varieties, in Algebraic Geometry, Sendai, 1985, ed. by T. Oda. Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), pp. 551–590

    Google Scholar 

  25. C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Complex Projective Spaces. Progress in Mathematics, vol. 3 (Birkhäuser, Boston, 1980)

    Google Scholar 

  26. B. Pasquier, N. Perrin, Local rigidity of quasi-regular varieties. Math. Z. 265(3), 589–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Th. Peternell, J. Wiśniewski, On stability of tangent bundles of Fano manifolds with b 2 = 1. J. Algebr. Geom. 4(2), 363–384 (1995)

    MATH  Google Scholar 

  28. D. Popovici, Limits of projective manifolds under holomorphic deformations. arXiv.AG/09102032v1

    Google Scholar 

  29. J. Potters, On almost homogeneous compact complex surfaces. Invent. math. 8, 244–268 (1968)

    Article  MathSciNet  Google Scholar 

  30. U. Schafft, Nichtsepariertheit instabiler Rang-2-Vektorbündel auf \(\mathbb{P}_{2}\). J. Reine Angew. Math. 338, 136–143 (1983)

    MathSciNet  MATH  Google Scholar 

  31. J. Wiśniewski, On Fano manifolds of large index. Manuscr. Math. 70, 145–152 (1991)

    Article  MATH  Google Scholar 

  32. J. Wiśniewski, On deformation of nef values. Duke Math. J. 64(2), 325–332 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Thomas Peternell & Florian Schrack

Authors
  1. Thomas Peternell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Florian Schrack
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Peternell .

Editor information

Editors and Affiliations

  1. Leibniz Universität Hannover, Hannover, Germany

    Anne Frühbis-Krüger

  2. Humboldt-University Berlin, Berlin, Germany

    Remke Nanne Kloosterman

  3. Leibniz Universität Hannover, Hannover, Germany

    Matthias Schütt

Additional information

Dedicated to Klaus Hulek on his 60th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Peternell, T., Schrack, F. (2014). Contact Kähler Manifolds: Symmetries and Deformations. In: Frühbis-Krüger, A., Kloosterman, R., Schütt, M. (eds) Algebraic and Complex Geometry. Springer Proceedings in Mathematics & Statistics, vol 71. Springer, Cham. https://doi.org/10.1007/978-3-319-05404-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation