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The Degree of Proper Holomorphic Mappings Between Special Domains in ℂn

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Abstract

Let Gi be closed Lie groups of U(n), Ωi be bounded Gi-invariant domains in ℂn which contains 0, and O(ℂn)Gi = ℂ, for i = 1, 2. It is known that if f: Ω1 →} Ω2 is a proper holomorphic mapping, and f−1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f.

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References

  1. Bell, S.: Proper holomorphic mappings and the Bergman projection. Duke Math. J., 48(1), 167–175 (1981)

    Article  MathSciNet  Google Scholar 

  2. Bell, S.: The Bergman kernel function and proper holomorphic mappings. Trans. Amer. Math. Soc., 270, 685–691 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Cartan, H.: Les fonctions de deux variables complexes et le problème de repr´esentation analytique. J. de Math. Pures et Appl., 96, 1–114 (1931)

    Google Scholar 

  4. Deng, F. S., Rong, F.: On biholomorphisms between bounded quasi-Reinhardt domains. Annali di Matematica, 195(3), 835–843 (2016)

    Article  MathSciNet  Google Scholar 

  5. Dinh, T. C., Sibony, N.: Equidistribution speed for endomorphisms of projective spaces. Math. Ann., 347(3), 613–626 (2010)

    Article  MathSciNet  Google Scholar 

  6. Heinzner, P.: On the automorphisms of special domains in ℂn. Indiana Univ. Math. J., 41, 707–712 (1992)

    Article  MathSciNet  Google Scholar 

  7. Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann., 289, 631–662 (1991)

    Article  MathSciNet  Google Scholar 

  8. Kaup, W.: Über das Randverhalten von Holomorphen Automorphismen beschränkter Gebiete. Manuscripta Math., 3, 250–270 (1970)

    MATH  Google Scholar 

  9. Narasimhan, R.: Several Complex Variables, University of Chicago Press, Chicago, 1971

    Google Scholar 

  10. Ning, J. F., Zhang, H. P., Zhou, X. Y.: Proper holomorphic mappings between invariant domains in ℂn. Trans. Amer. Math. Soc., 369, 517–536 (2017)

    MathSciNet  Google Scholar 

  11. Ning, J. F., Zhou, X. Y.: The degree of biholomorphic mappings between special domains in ℂn preserving 0. Sci. China Math., 60, 1077–1082 (2017)

    Article  MathSciNet  Google Scholar 

  12. Rong, F.: On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math., 140, 92–98 (2016)

    Article  MathSciNet  Google Scholar 

  13. Snow, D.: Reductive group action on Stein spaces. Math. Ann., 259, 79–97 (1982)

    Article  MathSciNet  Google Scholar 

  14. Yamamori, A.: Automorphisms of normal quasi-circular domains. Bull. Sci. Math., 138(3), 406–415 (2014)

    Article  MathSciNet  Google Scholar 

  15. Yamamori, A.: The linearity of origin-preserving automorphisms of quasi-circular domains. J. Math. Anal. Appl., 426(1), 612–623 (2015)

    Article  MathSciNet  Google Scholar 

  16. Yamamori, A., Zhang, L. Y.: On origin-preserving automorphisms of quasi-circular domains. J. Geom. Anal., 28(2), 1–13 (2017)

    MathSciNet  Google Scholar 

  17. Zhou, X. Y.: On orbit connectedness, orbit convexity, and envelopes of holomorphy. Izvestiya Russian Akad. Nauk, Series Math., 58(2), 196–205 (1994)

    MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Central South University, Changsha, 410083, P. R. China

    Jia Fu Ning

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

    Xiang Yu Zhou

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  1. Jia Fu Ning
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  2. Xiang Yu Zhou
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Correspondence to Jia Fu Ning or Xiang Yu Zhou.

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Supported by National Natural Science Foundation of China (Grant Nos. 11801572, 11688101)

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Ning, J.F., Zhou, X.Y. The Degree of Proper Holomorphic Mappings Between Special Domains in ℂn. Acta. Math. Sin.-English Ser. 36, 395–400 (2020). https://doi.org/10.1007/s10114-020-9299-z

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  • DOI: https://doi.org/10.1007/s10114-020-9299-z

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