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Inverse of Abelian integrals and ramified Riemann domains

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Abstract

We deal with the Levi problem (Hartogs’ inverse problem) for ramified Riemann domains by introducing a positive scalar function \(\rho (a, X)\) for a complex manifold X with a global frame of the holomorphic cotangent bundle by closed Abelian differentials, which is an analogue of Hartogs’ radius. We obtain some geometric conditions in terms of \(\rho (a, X)\) which imply the validity of the Levi problem for finitely sheeted ramified Riemann domains over \({\mathbf {C}}^n\). On the course, we give a new proof of the Behnke–Stein Theorem.

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Notes

  1. This fact was written twice in the introductions of his two papers, [25, 26]: The manuscript was written as a research report dated 12 Dec. 1943, sent to Teiji Takagi, then Professor at the Imperial University of Tokyo, and now one can find it in [29].

  2. It is noted that Oka VII [24] is different to his original, Oka VII in [27]; therefore, there are two versions of Oka VII. The English translation of Oka VII in [28] was taken from the latter, but unfortunately in [28] all the records of the received dates of the papers were deleted.

  3.  Cf. Definition 1.2.

References

  1. Andreotti, A., Narasimhan, R.: Oka’s Heftungslemma and the Levi problem for complex spaces. Trans. Am. Math. Soc. 111, 345–366 (1964)

    MathSciNet  MATH  Google Scholar 

  2. Behnke, H., Stein, K.: Entwicklung analytischer Funktionen auf Riemannschen Flächen. Math. Ann. 120, 430–461 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bers, L.: Introduction to Several Complex Variables, Lecture Notes. New York University, Courant Inst. Math. Sci. (1964)

  4. Cartan, H., Thullen, P.: Regularitäts- und Konvergenzbereiche. Math. Ann. 106, 617–647 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  5. Demailly, J.-P.: Complex Analytic and Differentiable Geometry (2012).www-fourier.ujf-grenoble.fr/ demailly/

  6. Elencwajg, G.: Pseudo-convexité locale dans les variétés kählériennes. Ann. Inst. Fourier Grenoble 25, 295–314 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fornæss, J.E.: A counterexample for the Levi problem for branched Riemann domains over \({\mathbf{C}}^n\). Math. Ann. 234, 275–277 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Forster, O.: Lectures on Riemann Surfaces, transl. by B. Gilligan, Grad. Texts Math. 81. Springer, New York (1981)

  9. Fritzsche, K., Grauert, H.: From Holomorphic Functions to Complex Manifolds, Grad. Texts Math. 213. Springer, New York (2002)

  10. Fujita, R.: Domaines sans point critique intérieur sur l’espace projectif complexe. J. Math. Soc. Jpn. 15, 443–473 (1963)

    Article  MATH  Google Scholar 

  11. Grauert, H.: Charakterisierung der holomorph vollständigen komplexen Räume. Math. Ann. 129, 233–259 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen Räumen. Math. Z. 65, 175–194 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grauert, H., Remmert, R.: Theorie der Steinschen Räume, Grundl. der Math. Wissen. vol. 227. Springer-Verlag, Berlin (1977): Translated to English by A. Huckleberry, Theory of Stein Spaces, Springer-Verlag, Berlin (1979): Translated to Japanese by K. Miyajima, Stein Kukan Ron, Springer, Tokyo (2009)

  14. Hörmander, L.: Introduction to Complex Analysis in Several Variables, First Edition 1966, Third Edition, North-Holland (1989)

  15. Kobayashi, S.: Hyperbolic Complex Spaces, Grundl. der Math. Wissen.  vol. 318. Springer, Berlin-New York (1998)

  16. Kusunoki, Y.: Function Theory (in Japanese). Asakura-Shoten, Tokyo (1973)

    Google Scholar 

  17. Lieb, I.: Le problème de Levi. Gazette Soc. Math. France 115, 9–34 (2008)

    MATH  Google Scholar 

  18. Narasimhan, R.: The Levi problem in the theory of several complex variables, ICM Stockholm (1962)

  19. Nishino, T.: Function Theory in Several Complex Variables (in Japanese), The University of Tokyo Press, Tokyo (1996): Translation into English by N. Levenberg and H. Yamaguchi, Amer. Math. Soc.  Providence, R.I. (2001)

  20. Noguchi, J.: Analytic Function Theory of Several Variables (in Japanese). Asakura-Shoten, Tokyo (2013): English translation in press, Springer, Singapore (2016)

  21. Noguchi, J., Winkelmann, J.: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation, Grundl. der Math. Wissen. vol. 350. Springer, Tokyo-Heidelberg-New York-Dordrecht-London (2014)

  22. Oka, K.: Sur les domaines pseudoconvexes. Proc. Imp. Acad. Tokyo 17, 7–10 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oka, K.: Sur les fonctions analytiques de plusieurs variables - VI Domaines pseudoconvexes. Tôhoku Math. J. 49, 15–52 (1942)

    MathSciNet  MATH  Google Scholar 

  24. Oka, K.: Sur les fonctions analytiques de plusieurs variables - VII Sur quelques notions arithmétiques. Bull. Soc. Math. France 78, 1–27 (1950)

    MathSciNet  MATH  Google Scholar 

  25. Oka, K.: Sur les fonctions analytiques de plusieurs variables - VIII Lemme fondamental. J. Math. Soc. Jpn. 3 (1), 204–214, (2) 259–278 (1951)

  26. Oka, K.: Sur les fonctions analytiques de plusieurs variables - IX Domaines finis sans point critique intérieur. Jpn. J. Math. 23, 97–155 (1953)

    MATH  Google Scholar 

  27. Oka, K.: Sur les fonctions analytiques de plusieurs variables. Iwanami Shoten, Tokyo (1961)

    MATH  Google Scholar 

  28. Oka, K.: Kiyoshi Oka Collected Papers, translated by R. Narasimhan, commentaries by H. Cartan, edited by R. Remmert, Springer-Verlag, Berlin-Tokyo (1984)

  29. Oka, K.: Kiyoshi Oka Digital Archieves, Library of Nara Women’s University. http://www.lib.nara-wu.ac.jp/oka/

  30. Schwartz, L.: Homomorphismes et applications complèment continues. C. R. l’Acad. Sci., Paris 236, 2472–2473 (1953)

  31. Serre, J.-P.: Deux théorèmes sur les applications complèment continues, Séminaire Henri Cartan 6, 1–7 (1953-1954)

  32. Takeuchi, A.: Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif. J. Math. Soc. Jpn. 16, 159–181 (1964)

    Article  MATH  Google Scholar 

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Acknowledgments

The author is very grateful to Professor J.E. Fornæss for the clarification that his example ([7]) does not satisfy Cond A (Sect. 3 (a)), and to Professor Makoto Abe for interesting discussions on the present theme.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Junjiro Noguchi

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  1. Junjiro Noguchi
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Correspondence to Junjiro Noguchi.

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Research supported in part by Grant-in-Aid for Scientific Research (C) 15K04917.

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Noguchi, J. Inverse of Abelian integrals and ramified Riemann domains. Math. Ann. 367, 229–249 (2017). https://doi.org/10.1007/s00208-016-1384-3

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