×

The Kobayashi metric on complex spaces. (English) Zbl 0859.32011

A program is carried out in the paper under review to extend the notion of Kobayashi-Royden pseudometric to complex analytic spaces and to prove the analogue of the fact that the Kobayashi pseudodistance is the integrated form of the Kobayashi-Royden pseudometric. Given a connected complex analytic space \(X\), a family \(\{K^k_X\}\) of Kobayashi infinitesimal pseudometrics, depending on the positive integer \(k\), is introduced, acting on the bundle of jets of order \(k\) of holomorphic curves in \(X\). (When \(X\) is a smooth complex manifold, then \(K^1_X\) is the usual Kobayashi-Royden pseudometric.) It is shown that the Kobayashi pseudodistance is the integrated form of \(K^k_X\).

MSC:

32F45 Invariant metrics and pseudodistances in several complex variables
32C15 Complex spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Mediterrean Press, Cosenza, 1989 · Zbl 0747.32002
[2] Do Duc Thai: Royden-Kobayashi pseudometric and tautness of normalization of complex spaces. Boll. U.M.I., VIII, Ser. A5, No. 2, (1991), 147-156 · Zbl 0742.32018
[3] Fisher, G.: Complex Analytic Geometry. L.N.M. no. 538, Springer-Verlag, 1976
[4] Franzoni, T., Vesentini, ?.: Holomorphic maps and invariant distances. North Holland, 1972 · Zbl 0447.46040
[5] Grauert, H.: Jetmetriken und hyperbolische Geometrie. Math. Z.200 (2) (1989) 149-168 · Zbl 0664.32020 · doi:10.1007/BF01230277
[6] Grothendieck, A.: Techniques de construction en g?om?trie analytique. S?minaire Henri Cartan, 13i?me ann?e, (1960/61)
[7] Harris, L.A.: Schwartz-Pick systems of pseudometrics for domains in normed linear spaces. In: ?Advances in Holomorphy?, (Barroso ed.). Notas de Matematica 65. North Holland, Amsterdam, 1979, pp. 345-406
[8] Hironaka, H.: Resolution of an algebraic variety over a field of characteristic zero I?II. Ann. of Math.79 (2) (1964) pp. 109-203, 205-326 · Zbl 0122.38603 · doi:10.2307/1970486
[9] Kobayashi, S.: Invariant distances on complex manifolds and holomorphic mappings. Math. Soc. Japan19 (1967) 460-480 · Zbl 0158.33201 · doi:10.2969/jmsj/01940460
[10] Kobayashi, S.: Distance, holomorphic mappings and the Schwarz lemma. Math. Soc. Japan19 (1967), 481-485 · Zbl 0158.33202 · doi:10.2969/jmsj/01940481
[11] Kobayashi, S.: Hyperbolic manifolds and holomorphic mappings. Dekker, New York, 1970 · Zbl 0207.37902
[12] Kobayashi, S.: Intrinsic distances, measures and geometric function theory. Bull. Amer. Math. Soc.82 (1976) 357-416 · Zbl 0346.32031 · doi:10.1090/S0002-9904-1976-14018-9
[13] Kobayashi, S.: A new invariant infinitesimal metric. Int. J. Math.1 (1990) 83-90 · Zbl 0702.32021 · doi:10.1142/S0129167X9000006X
[14] Lang, S.: An introduction to complex hyperbolic spaces. Springer, New York, 1987 · Zbl 0628.32001
[15] ?ojasiewicz, S.: Ensembles semi-analytiques. I.H.E.S., Bures-sur-Yvette, 1965 · Zbl 0241.32005
[16] Royden, H.: Remarks on the Kobayashi metric. In: Several Complex Variables II. Lect. Notes in Math.189. Springer, Berlin, 1971, 125-137 · Zbl 0218.32012
[17] Royden, H.: The extension of regular holomorphic maps. Proc. Amer. Math. Soc.43 (1974) 306-310 · Zbl 0292.32019 · doi:10.1090/S0002-9939-1974-0335851-X
[18] Venturini, S.: Pseudodistances and pseudometrics on real and complex manifolds. Ann. Mat. Pura e Appl., Serie (IV)154 (1989) 385-402 · Zbl 0702.32022 · doi:10.1007/BF01790358
[19] Withney, H.: Complex Analytic Varieties. Addison Wesley, Reading, Mass., 1972
[20] Wu, H.: Old and new invariant metrics. In: Fornaess, J.E. (ed) Proc. of the Mittag-Leffler Institute, 1987-1988, Math. Notes38 1993, Princeton, pp. 640-682 · Zbl 0773.32017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.