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Some results about holomorphic vector bundles over general Hopf manifolds

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Abstract

In the present note that grew out of my talk given at the conference in honor of Prof. Zhong Tongde, I give a survey of some recent results about holomorphic vector bundles over general Hopf manifolds.

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References

  1. Calabi E, Eckmann B. A class of compact, complex manifolds which are not algebraic. Ann of Math, 58(2): 494–500 (1953)

    Article  MathSciNet  Google Scholar 

  2. Kobayashi S. Differential geometry of complex vector bundles. Princeton: Princeton University Press, 1987

    MATH  Google Scholar 

  3. Zhou X Y. A remark on the Douady sequence for non-primary Hopf manifolds. Asian J Math, 8: 131–136 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Gan N, Zhou X Y. The cohomology of vector bundles on general non-primary Hopf manifolds. In: Contemporary Mathematics, Vol. 400, Providence, RI: American Mathematical Society, 2006, 107–116

    Google Scholar 

  5. Atiyah M. Complex analytic connections in fibre bundles. Trans Amer Math Soc, 85: 181–207 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu W M, Zhou X Y. Line bundles on non-primary Hopf manifolds. Sci China Ser A, 47: 538–551 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mall D. On holomorphic vector bundle on Hopf Manifolds with pullback on the universal covering. Math Ann, 294: 719–740 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mall D. Filtrable bundles of rank 2 on Hopf surfaces. Math Z, 212: 239–256 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mall D. The cohomology of line bundles on Hopf Manifolds. Osaka J Math, 28: 999–1015 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Mall D. Contractions, Fredholm operators and the cohomology of vector bundles on Hopf manifolds. Arch Math, 66: 71–76 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gan N, Zhou X Y. The structure of Vector bundles on non-primary Hopf manifolds. Preprint

  12. Zhou X Y, Liu W M. Holomorphic vector bundles on Hopf manifolds with Abelian fundamental groups. Acta Mathematica Sinica, English Series, 20: 605–612 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mall D. Hopf manifolds and vector bundles on projective spaces. Abh Math Sem Univ Hamburg, 64: 163–166 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gan N, Zhou X Y. The cohomology of line bundles on non-primary Hopf manifolds. Acta Mathematica Sinica, English Series, 23: 289–296 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Borcea C. Some remarks on deformations of Hopf manifolds. Rev Roum Math Pures et Appl, 26: 1287–1294 (1981)

    MATH  MathSciNet  Google Scholar 

  16. Brînzǎnescu V. Holomorphic vector bundles over compact complex surfaces. In: Lecture Notes in Mathematics, Vol. 1624. Berlin: Springer-Verlag, 1996

    Google Scholar 

  17. Brinzanescu V, Flondor P. Bundles on nonalgebraic 2-tori. J Reine Angew Math, 363: 47–58 (1985)

    MATH  MathSciNet  Google Scholar 

  18. Dabrowski K. Moduli spaces for Hopf surfaces. Math Ann, 259: 201–225 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  19. Douady A, Seminare H. Cartan. Expose 3, 1960/1961

  20. Elencwaj G, Foster O. Vector bundles on manifolds without divisors and a Theorem on deformation. Ann L’Inst Fourier, 32: 25–51 (1982)

    Google Scholar 

  21. Fritzsche K, Grauert H. From holomorphic functions to complex manifolds. In: Graduate Texts in Mathematics, Vol. 213. New York: Springer-Verlag, 2002

    MATH  Google Scholar 

  22. Kato M. Topology of Hopf surfaces. J Math Soc Japan, 27: 222–238 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kodaira K. On the structure of compact complex analytic surfaces. Amer J Math, 86: 751–798 (1964); 88: 682–721 (1966); 90: 55–83 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  24. Matumoto T, Nakagawa N. Explicit description of Hopf surfaces and their automorphism groups. Osaka J Math, 37: 417–424 (2000)

    MATH  MathSciNet  Google Scholar 

  25. Schwarzenberger R. Vector bundles on algebraic surfaces. Proc London Math Soc, 11: 601–622 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wehler J. Versal deformation of Hopf surfaces. J Reine Angew Math, 328: 22–32 (1981)

    MATH  MathSciNet  Google Scholar 

  27. Wu W T. Sur les Espace Fibrés. In: Act Sci Ind, Vol. 1183. Paris: Hermann, 1952

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China

    XiangYu Zhou

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  1. XiangYu Zhou
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Correspondence to XiangYu Zhou.

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Dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday

This work was supported by National Science Foundation of China (Grant Nos. 10421101, 10721061)

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Zhou, X. Some results about holomorphic vector bundles over general Hopf manifolds. Sci. China Ser. A-Math. 52, 2863–2866 (2009). https://doi.org/10.1007/s11425-009-0223-3

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  • DOI: https://doi.org/10.1007/s11425-009-0223-3

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