Advertisement

Several injectivity theorems on compact Kähler manifolds

  • Chunle HuangEmail author
Original Paper
  • 12 Downloads

Abstract

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kähler manifolds, which generalize Enoki’s original injectivity theorem.

Keywords

Injectivity theorem Compact Kähler manifolds Bochner technique Hodge theory 

Mathematics Subject Classification (2000)

32L10 32Q15 

Notes

Acknowledgements

The author would like to thank the referee for carefully reading the paper and for valuable suggestions.

References

  1. 1.
    Ambro, F.: An injectivity theorem. Compos. Math. 150, 999–1023 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Demailly, J.P.: Complex analytic and differential geometry. Université de Grenoble I, Grenoble (1997) Google Scholar
  3. 3.
    Demailly, J.P.: Estimations \({\rm L}^2\) pour l’oprateur \(\bar{\partial }\) d’un fibr vectoriel holomorphe semi-positif au-dessus d’une varit kählrienne complte. Ann. Sci. l’Ecole Norm. Suprieure 15(3), 457–511 (1982)CrossRefGoogle Scholar
  4. 4.
    Enoki, I.: Kawamata-Viehweg Vanishing Theorem for Compact Kähler Manifolds, Einstein Metrics and Yang–Mills Connections (Sanda, 1990). Lecture Notes in Pure and Applied Mathematics, vol. 145, pp. 59–68. Dekker, New York (1993)Google Scholar
  5. 5.
    Fujino, O.: A transcendental approach to Koll’s injectivity theorem. Osaka J. Math. 49(3), 833–852 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fujino, O.: On semipositivity, injectivity, and vanishing theorems, preprint (2015). arXiv:1503.06503v3
  7. 7.
    Fujino, O.: Enoki’s injectivity theorem (Private note) (2011)Google Scholar
  8. 8.
    Fujino, O., Matsumura, S.: Injectivity theorem for pseudo-effective line bundles and its applications. arXiv preprint arXiv:1605.02284 (2016)
  9. 9.
    Huang, C.: Injectivity theorems on compact complex manifolds. Sci. China Math. 61(61), 1089 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math. (2) 123(1), 11–42 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math. (2) 124, 171–202 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Matsumura, S.: An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. arXiv preprint arXiv:1308.2033 (2013)
  13. 13.
    Matsumura, S.: Injectivity theorems with multiplier ideal sheaves and their applications. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, K.-T., Shcherbina, N. (eds.) Complex Analysis and Geometry, pp. 241–255. Springer, Tokyo (2015)CrossRefGoogle Scholar
  14. 14.
    Matsumura, S.: A transcendental approach to injectivity theorem for log canonical pairs. arXiv preprint arXiv:1607.07213 (2016)

Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Institute of MathematicsHunan UniversityChangshaChina

Personalised recommendations