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Injectivity theorems on compact complex manifolds

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Abstract

We use analytic methods in this paper to prove some new Enoki type injectivity theorems on compact complex manifolds which generalize more or less the original Enoki injectivity theorem.

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References

  1. Ambro F. An injectivity theorem. Compos Math, 2014, 150: 999–1023

    Article  MathSciNet  MATH  Google Scholar 

  2. Demailly J P. L 2 vanishing theorems for positive line bundles and adjunction theory. In: Transcendental Methods in Algebraic Geometry. Berlin-Heidelberg: Springer, 1996, 1–97

    Chapter  Google Scholar 

  3. Demailly J P. Analytic Methods in Algebraic Geometry. Surveys of Modern Mathematics, vol. 1. Beijing: Higher Education Press, 2010

  4. Demailly J P. Complex Analytic and Differential Geometry. Http://www-fourier.ujf-grenoble.fr/~sdemailly/books, 2017

    Google Scholar 

  5. Demailly J P, Peternell T, Schneider M. Pseudo-effective line bundles on compact Kähler manifolds. Internat J Math, 2001, 12: 689–741

    Article  MathSciNet  MATH  Google Scholar 

  6. Enoki I. Kawamata-Viehweg vanishing theorem for compact Kähler manifolds. In: Einstein Metrics and Yang-Mills Connections. Lecture Notes in Pure and Applied Mathematics, vol. 145. New York: Dekker, 1993, 59–68

    MathSciNet  MATH  Google Scholar 

  7. Esnault H, Viehweg E. Logarithmic de Rham complexes and vanishing theorems. Invent Math, 1986, 86: 161–194

    Article  MathSciNet  MATH  Google Scholar 

  8. Esnault H, Viehweg E. Lectures on Vanishing Theorems. Basel: Birkhäuser, 1992

    Book  MATH  Google Scholar 

  9. Fujiki A. An L 2 Dolbeault lemma and its applications. Publ Res Inst Math Sci, 1992, 28: 845–884

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujino O. Enoki’s injectivity theorem. Https://www.math.kyoto-u.ac.jp/sfujino/enoki-inj.pdf, 2011

    Google Scholar 

  11. Fujino O. On semipositivity, injectivity, and vanishing theorems. ArXiv:1503.06503v3, 2015

    MATH  Google Scholar 

  12. Fujino O, Matsumura S. Injectivity theorem for pseudo-effective line bundles and its applications. ArXiv:1605.02284, 2016

    Google Scholar 

  13. Hörmander L. L 2 estimates and existence theorems for the \(\bar \partial \) operator. Acta Math, 1965, 113: 89–152

    Article  MathSciNet  MATH  Google Scholar 

  14. Kollár J. Higher direct images of dualizing sheaves II. Ann of Math (2), 1986, 123: 11–42

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu K F, Sun X F, Yang X K. Positivity and vanishing theorems for ample vector bundles. J Algebraic Geom, 2013, 22: 303–331

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo H Z. Stability of algebraic manifolds. PhD Thesis. Cambridge: Massachusetts Institute of Technology, 1998

    Google Scholar 

  17. Manivel L. Vanishing theorems for ample vector bundles. Invent Math, 1997, 127: 401–416

    Article  MathSciNet  MATH  Google Scholar 

  18. Matsuki K, Olsson M. Kawamata-Viehweg vanishing as Kodaira vanishing for stacks. Math Res Lett, 2005, 12: 207–217

    Article  MathSciNet  MATH  Google Scholar 

  19. Matsumura S. A transcendental approach to injectivity theorem for log canonical pairs. ArXiv:1607.07213, 2016

    Google Scholar 

  20. Matsumura S. An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. J Algebraic Geom, 2017, in press

    Google Scholar 

  21. Morrow J A, Kodaira K. Complex Manifolds. Providence: Amer Math Soc, 1971

    MATH  Google Scholar 

  22. Norimatsu Y. Kodaira vanishing theorem and Chern classes for ϑ-manifolds. Proc Japan Acad Ser A Math Sci, 1978, 54: 107–108

    Article  MathSciNet  MATH  Google Scholar 

  23. Shiffman B, Sommese A J. Vanishing Theorems on Complex Manifolds. Progress in Mathematics, vol. 56. Boston: Birkhäuser, 1985

  24. Zhao C. An L 2 injectivity theorem and its application. Pure Appl Math Q, 2015, 11: 369–392

    Article  MathSciNet  MATH  Google Scholar 

  25. Zucker S. Hodge theory with degenerating coefficients: L 2 cohomology in the Poincaré metric. Ann of Math (2), 1979, 109: 415–476

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author expresses his sincere gratitude to Prof. Kefeng Liu for his useful advice and interest on this work.

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Correspondence to Chunle Huang.

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Huang, C. Injectivity theorems on compact complex manifolds. Sci. China Math. 61, 1089–1098 (2018). https://doi.org/10.1007/s11425-016-9090-5

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  • DOI: https://doi.org/10.1007/s11425-016-9090-5

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