Skip to main content
SpringerLink
Log in

Generically nef vector bundles and geometric applications

  • Conference paper
  • First Online:
Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. We show that the cotangent bundle is actually generically ample, if the manifold is of general type and study in detail the case of intermediate Kodaira dimension. Moreover, manifolds with generically nef and ample tangent bundles are investigated as well as connections to classical theorems on vector fields on projective manifolds.

Mathematics Subject Classification (2010) 14E15, 14J40, 14E30, 14M22.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barth, W.P.; Hulek, K.; Peters, C.A.M.; Van de Ven, A.: Compact complex surfaces.Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 2004.

    Google Scholar 

  2. Birkar,C.; Cascini,P.; Hacon,D.; McKernan,J.: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), 405-468.

    Google Scholar 

  3. Boucksom,S.; Demailly,J.P.; Paun,M.; Peternell,Th.: The pseudo-effective cone of a compact K¨ahler manifold and varieties of negative Kodaira dimension. Preprint 2004

    Google Scholar 

  4. Bogomolov,F.; McQuillan,M.: Rational curves on foliated varieties. Preprint, IHES, 2001.

    Google Scholar 

  5. Bogomolov,F.: Holomorphic tensors and vector bundles on projective varieties. Math.USSR Izv. 13 (1979), 499-555

    Google Scholar 

  6. Berndtsson,B.; Paun,M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145 (2008), no. 2, 341–378.

    Google Scholar 

  7. Campana,F.; Peternell,Th.: Rational curves and ampleness properties of the tangent bundle of algebraic varieties. Manuscripta Math. 97 (1998), no. 1, 59–74.

    Google Scholar 

  8. Campana,F.; Peternell, Th.: Geometric stability of the cotangent bundle and the universal cover of a projective manifold (with an appendix by Matei Toma). arXiv:math/0405093; to appear in Bull. Soc. Math. France

    Google Scholar 

  9. Demailly,J.P.; Peternell,Th.; Schneider,M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebr. Geom. 3, No.2, 295-345 (1994).

    Google Scholar 

  10. Demailly J.-P., Peternell Th., Schneider M.: Holomorphic line bundles with partially vanishing cohomology. Conf. in honor of F. Hirzebruch, Israel Mathematical Conference Proceedings Vol. 9 (1996) 165-198

    Google Scholar 

  11. Demailly, J.P.; Peternell, Th.; Schneider, M.: Pseudo-effective line bundles on compact K¨ahler manifolds. Int. J. Math. 12, No. 6, 689-741 (2001).

    Google Scholar 

  12. Demailly,J.P., Peternell,Th.: The movable cone - an example. Unpublished note (2007)

    Google Scholar 

  13. Enoki,I.: Stability and negativity for tangent sheaves of minimal Khler spaces. Geometry and analysis on manifolds (Katata/Kyoto, 1987), 118–126, Lecture Notes in Math., 1339, Springer, Berlin, 1988.

    Google Scholar 

  14. Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds. Demailly, J.P. (ed.) et al., School on vanishing theorems and effective results in algebraic geometry. ICTP Lect. Notes 6, 335-393 (2001).

    Google Scholar 

  15. Kawamata,Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79 (1985), no. 3, 567–588.

    Google Scholar 

  16. Kobayashi,S.: Differential geometry on complex vector bundles. Iwanami Shoten and Princeton Univ. Press, 1987

    Google Scholar 

  17. Koll´ar.J.: Flop. Nagoya Math. J. 113, 15–36 (1989)

    Google Scholar 

  18. Koll´ar,J.: Rational curves on algebraic varieties. Springer (1996)

    Google Scholar 

  19. Kebekus,S.; Sola Cond´e,L; Toma,M.: Rationally connected foliations after Bogomolov and McQuillan. J. Algebraic Geom. 16 (2007), no. 1, 65–81.

    Google Scholar 

  20. Lieberman,D.: Compactness of the Chow scheme: applications to automorphisms and deformations of K¨ahler manifolds. Lect. Notes in Math. 670, 14–186. Springer 1978

    Google Scholar 

  21. Miyaoka,Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Sendai, 1985, 449–476, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987.

    Google Scholar 

  22. Miyaoka,Y.: On the Kodaira dimension of minimal threefolds. Math. Ann. 281 (1988), no. 2, 325–332.

    Google Scholar 

  23. Miyaoka,Y; Mori,S.: A numerical criterion for uniruledness. Ann. Math. (2) 124, 65-69 (1986).

    Google Scholar 

  24. Miyaoka,Y.; Peternell,T.: Geometry of higher dimensional varieties. DMV seminar, vol. 26. Birkh¨auser 1997

    Google Scholar 

  25. Mehta,V.B.; Ramanathan,A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258, 213–224 (1982)

    Google Scholar 

  26. Okonek,C.; Schneider,M.; Spindler,H.: Vector bundles on complex projective spaces. Progress in Mathematics, 3. Birkhuser, Boston, Mass., 1980.

    Google Scholar 

  27. Peternell, Th.: Kodaira dimension of subvarieties. II. Int. J. Math. 17, No. 5, 619-631 (2006).

    Google Scholar 

  28. Peternell,Th.: Varieties with generically nef tangent bundles. arXiv:0807.0982

    Google Scholar 

  29. Peternell,Th.; Serrano, F.: Threefolds with nef anticanonical bundles. Collect. Math. 49, No.2-3, 465–517 (1998).

    Google Scholar 

  30. Peternell,Th.; Sommese,A.J.: Ample vector bundles and branched coverings. With an appendix by Robert Lazarsfeld. Special issue in honor of Robin Hartshorne. Comm. Algebra 28 (2000), no. 12, 5573–5599.

    Google Scholar 

  31. Peternell,Th.; Sommese,A.J.: Ample vector bundles and branched coverings. II. The Fano Conference, 625–645, Univ. Torino, Turin, 2004.

    Google Scholar 

  32. Peternell,Th.; Wisniewski,J.: On stability of tangent bundles of Fano manifolds with b2 = 1: J. Alg. Geom. 4, 363–384 (1995)

    Google Scholar 

  33. Shepherd-Barron,N.: Miyaoka’s theorem on the seminegativity of TX : In: Flips an abundance for algebraic threefolds; ed. J. Koll´ar; Ast´erisque 211, 103-114 (1992)

    Google Scholar 

  34. Shepherd-Barron,N.: Semi-stability and reduction mod p. Topology 37, 659-664 (1998) [Ts88] Tsuji,H.: Stability of tangent bundles of minimal algebraic varieties. Topology 27 (1988), no. 4, 429–442.

    Google Scholar 

  35. Zhang,Q.: On projective manifolds with nef anticanonical divisors. Math. Ann. 332 (2005), 697–703

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

    Thomas Peternell

Authors
  1. Thomas Peternell
    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. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Peternell, T. (2011). Generically nef vector bundles and geometric applications. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_17

Download citation

Publish with us

Policies and ethics

Search

Navigation