Skip to main content
SpringerLink
Log in

Logarithmic vanishing theorems for effective q-ample divisors

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Let X be a compact Kähler manifold and D be a simple normal crossing divisor. If D is the support of some effective q-ample divisor, we show

$$H^{i}(X, \Omega_{X}^{j}(\log D))=0, \quad \text { for } \quad i+j>n+q.$$

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andreotti A, Grauert H. Théorè me de finitude pour la cohomologie des espaces complexes. Bull Soc Math France, 1962, 90: 193–259

    Article  MathSciNet  Google Scholar 

  2. Broomhead N, Ottem J-C, Prendergast-Smith A. Partially ample line bundles on toric varieties. Glasg Math J, 2016, 58: 587–598

    Article  MathSciNet  Google Scholar 

  3. Deligne P. Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ Math Inst Hautes Études Sci, 1969, 35: 107–126

    Article  Google Scholar 

  4. Deligne P. Théorie de Hodge II. Publ Math Inst Hautes Études Sci, 1972, 40: 5–57

    Article  Google Scholar 

  5. Demailly J-P. A converse to the Andreotti-Grauert theorem. Ann Fac Sci Toulouse Math (6), 2011, 20: 123–135

    Article  MathSciNet  Google Scholar 

  6. Demailly J-P, Peternell T, Schneider M. Holomorphic line bundles with partially vanishing cohomology. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry. Israel Mathematical Conference Proceedings, vol. 9. Ramat Gan: Bar-Ilan University, 1996, 165–198

    Google Scholar 

  7. Girbau J. Sur le théorè me de Le Potier d’annulation de la cohomologie. C R Acad Sci Paris Sér A, 1976, 283: 355–358

    MathSciNet  MATH  Google Scholar 

  8. Greb D, Küronya A. Partial positivity: Geometry and cohomology of q-ample line bundles. In: Recent Advances in Algebraic Geometry. London Mathematical Society Lecture Note Series, vol. 417. Cambridge: Cambridge University Press, 2015, 207–239

    Chapter  Google Scholar 

  9. Griffith P, Harris J. Principles of Algebraic Geometry. New York: Wiley, 1978

    Google Scholar 

  10. Huang C, Liu K, Wan X, et al. Logarithmic vanishing theorems on compact Kähler manifolds I. ArXiv:1611.07671, 2016

  11. Matsumura S. Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on a surface. Ann Inst Fourier (Grenoble), 2013, 63: 2199–2221

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Shiftman B, Sommese A J. Vanishing Theorems on Complex Manifolds. Progress in Mathematics, vol. 56. Boston: Birkhäauser, 1985

    Book  Google Scholar 

  14. Yang X. A partial converse to the Andreotti-Grauert theorem. Compos Math, 2019, 155: 89–99

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences Research Center, Chongqing University of Science and Technology, Chongqing, 400054, China

    Kefeng Liu

  2. Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, 90095, USA

    Kefeng Liu

  3. School of Mathematics, Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

    Xueyuan Wan

  4. Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Xiaokui Yang

Authors
  1. Kefeng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xueyuan Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaokui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kefeng Liu.

Additional information

Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, K., Wan, X. & Yang, X. Logarithmic vanishing theorems for effective q-ample divisors. Sci. China Math. 62, 2331–2334 (2019). https://doi.org/10.1007/s11425-019-9553-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9553-2

Keywords

MSC(2010)

Search

Navigation