Abstract
The purpose of this article is threefold. In 1971, H. L. Royden introduced the Kobayashi-Royden pseudometrics on (nonsingular) complex manifolds and gave a criterion for the Kobayashi hyperbolicity of (nonsingular) complex manifolds which plays an essentially important role in Hyperbolic Complex Geometry. Perhaps, since there are some unresolved technical problems, the above theorem of Royden is not yet completely cleared up for singular complex spaces. Thus, the first is to prove Royden’s theorem for singular complex spaces. The second is to give a criteria of the hyperbolicity modulo a closed subset of singular complex spaces from the viewpoint of comparing the Kobayashi k-differential pseudometrics on these spaces and the Landau property of the ones. The third is to investigate the hyperbolicity modulo a closed subset of domains in a singular complex space from the condition on localization at their boundary points.
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References
Berteloot, F.: Attraction des disques analytiques et continuité höldérienne d’applications holomorphes propres. Topics in Complex Analysis. Banach Center Publ. 31, 91–98 (1995)
Fornaess, J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248, 47–72 (1980)
Gaussier, H.: Tautness and complete hyperbolicity of domains in \(\mathbb {C}^{n}\). Proc. Am. Math. Soc. 127, 105–116 (1999)
Hanh, T.K., Kim, T.K.: Hyperbolicity of a complex manifold and other equivalent properties. Proc. Am. Math. Soc. 91, 49–53 (1984)
Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren der mathematischen Wissenschaften, vol. 318 (1998)
Lang, S.: Introduction to Complex Hyperbolic Spaces. Springer-Verlag, New York (1987)
Peternell, Th.: Pseudoconvexity, the Levi problem and vanishing theorems. Encycl. Math. Sci. Several Complex Variables VII 74, 221–257 (1994)
Royden, H.L.: Remarks on the Kobayashi Metric. Lecture Notes in Math., vol. 185, pp 125–137. Springer, New York (1971)
Thai, D.D.: Royden-Kobayashi pseudometric and tautness of normalizations of complex spaces. Bollettino. U. M. I. 5-A(7), 147–156 (1991)
Thai, D.D., Duc, P.V.: The Kobayashi k-metrics on complex spaces. Inter. J. Math. 10, 917–924 (1999)
Thai, D.D., Duc, P.V.: On the complete hyperbolicity and the tautness of the Hartogs domains. Inter. J. Math. 11, 103–111 (2000)
Trao, N.V., Minh, T.H.: Remarks on the Kobayashi hyperbolicity of complex spaces. Acta Math. Vietnam. 34, 375–387 (2009)
Venturini, S.: The Kobayashi metric on complex spaces. Math. Ann. 305, 25–34 (1996)
Acknowledgements
This work was done during a stay of the first author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). We would like to thank VIASM for partial support, and the staff of VIASM for their hospitality.
The research of the authors is supported by an NAFOSTED grant of Vietnam (Grant No. 101.04-2017.317).
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Trang, P.N.T., Van Trao, N. On Hyperbolicity Modulo a Closed Subset of Singular Complex Spaces. Acta Math Vietnam 47, 719–729 (2022). https://doi.org/10.1007/s40306-021-00462-x
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DOI: https://doi.org/10.1007/s40306-021-00462-x