Grothendieck–Lefschetz for ample subvarieties

Mathematische Zeitschrift (2021)Cite this article

Abstract

We establish a Grothendieck–Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. 1.

    Achinger, P., Witaszek, J., Zdanowicz, M.: Liftability of the Frobenius morphism and images of toric varieties, arXiv e-prints (2017), available at arXiv:1708.03777

  2. 2.

    Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962) (French)

  3. 3.

    Arapura, D.: Frobenius amplitude and strong vanishing theorems for vector bundles. Duke Math. J. 121(2), 231–267 (2004). With an appendix by Dennis S. Keeler

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bădescu, L: Projective geometry and formal geometry, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 65, Birkhäuser Verlag, Basel, (2004)

  5. 5.

    Bădescu, Lucian, Schneider, Michael: A criterion for extending meromorphic functions. Math. Ann. 305(2), 393–402 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bădescu, L., Schneider, M.: Formal functions, connectivity and homogeneous spaces, Algebraic geometry. de Gruyter, Berlin (2002)

  7. 7.

    Beltrametti, M. C., Ionescu, P.: A view on extending morphisms from ample divisors, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, pp. 71–110 (2009)

  8. 8.

    Debarre, O.: Fulton-Hansen and Barth–Lefschetz theorems for subvarieties of abelian varieties. J. Reine Angew. Math. 467, 187–197 (1995)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Debarre, O.: On the geometry of abelian varieties, Lecture notes (2008)

  10. 10.

    de Fernex, T., Küronya, A., Lazarsfeld, R.: Higher cohomology of divisors on a projective variety. Math. Ann. 337(2), 443–455 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    de Fernex, T., Lau, C.C.: Extending rationally connected fibrations from ample subvarieties (2020). Preprint, arXiv:2011.10567

  12. 12.

    Demailly, J.-P.: A converse to the Andreotti-Grauert theorem (2011). Preprint, arXiv: 1101.3635

  13. 13.

    Demailly, J.-P., Peternell, T., Schneider, M.: Holomorphic line bundles with partially vanishing cohomology. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, (1996)

  14. 14.

    Gieseker, D.: On two theorems of Griffiths about embeddings with ample normal bundle. Am. J. Math. 99(6), 1137–1150 (1977)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Grothendieck, A.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposè par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2

  16. 16.

    Halic, M.: Subvarieties with partially ample normal bundle. Math. Z. 293(1–2), 371–382 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hartshorne, Robin: Cohomological dimension of algebraic varieties. Ann. Math. (2) 88, 403–450 (1968)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, vol. 156. Springer-Verlag, Berlin-New York (1970)

  19. 19.

    Hironaka, H.: On some formal imbeddings. Ill. J. Math. 12, 587–602 (1968)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hironaka, H., Matsumura, H.: Formal functions and formal embeddings. J. Math. Soc. Jpn 20, 52–82 (1968)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lau, Chung-Ching: Numerical dimension and locally ample curves. Doc. Math. 23, 677–696 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Lau, Chung Ching: On nef subvarieties. Adv. Math. 353, 396–430 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Lazarsfeld, R.: Positivity in algebraic geometry. I and II, 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-Verlag, Berlin, (2004)

  24. 24.

    Litt, D.: Vanishing for Frobenius Twists of Ample Vector Bundles, arXiv e-prints (2017), available at arXiv:1702.03962

  25. 25.

    Litt, D.: Non-Abelian Lefschetz hyperplane theorems. J. Algebraic Geom. 27(4), 593–646 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Manivel, L.: Théorèmes d’annulation sur certaines variétés projectives, Comment. Math. Helv. 71, no. 3, 402–425 (1996) (French)

  27. 27.

    Vikram, B., Mehta, V.B., Srinivas, V.: Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64(2), 191–212 (1987). With an appendix by Srinivas and M. V. Nori

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Mustaţă, M., Srinivas, V.: Ordinary varieties and the comparison between multiplier ideals and test ideals. Nagoya Math. J. 204, 125–157 (2011)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Ottem, J.C.: Ample subvarieties and q-ample divisors. Adv. Math. 229(5), 2868–2887 (2012)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Sommese, A.J.: On manifolds that cannot be ample divisors. Math. Ann. 221(1), 55–72 (1976)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Sommese, A.J.: Submanifolds of Abelian varieties. Math. Ann. 233(3), 229–256 (1978)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Speiser, R.: Cohomological dimension and Abelian varieties. Am. J. Math. 95, 1–34 (1973)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Speiser, R.: Formal meromorphic functions and cohomology on an algebraic variety. Nagoya Math. J. 77, 125–135 (1980)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Totaro, B.: Line bundles with partially vanishing cohomology. J. Eur. Math. Soc. (JEMS) 15(3), 731–754 (2013)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We thank Daniel Litt for discussions on Sommese’s conjecture in relation to his work [25] and Adrian Langer for useful comments. We also thank the referee for useful comments and suggestions.

Author information

Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Tommaso de Fernex

  2. Hong Kong, China

    Chung Ching Lau

Authors
  1. Tommaso de Fernex
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chung Ching Lau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso de Fernex.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of the first author was partially supported by NSF Grant DMS-1700769 and by NSF Grant DMS-1440140 while in residence at MSRI in Berkeley during the Spring 2019 semester. The research of the second author was partially supported by a Croucher Foundation Fellowship .

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

de Fernex, T., Lau, C.C. Grothendieck–Lefschetz for ample subvarieties. Math. Z. (2021). https://doi.org/10.1007/s00209-020-02693-4

Download citation

Keywords

  • Ample subvariety
  • Picard group
  • Abelian variety

Mathematics Subject Classification

  • Primary 14C22
  • Secondary 14K12
  • 14F17