Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Tranversality Theorem
for Holomorphic Mappings and
Stability of Eisenman-Kobayashi Measures


Authors: Sh. Kaliman and M. Zaidenberg
Journal: Trans. Amer. Math. Soc. 348 (1996), 661-672
MSC (1991): Primary 32E10, 32H02, 58C10, 58A35, 58A07
DOI: https://doi.org/10.1090/S0002-9947-96-01482-1
MathSciNet review: 1321580
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that Thom's Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping $f:S\rightarrow M$ from a Stein manifold $S$ to a complex manifold $M$ and given an analytic subset $A$ of the jet space $J^{k} (S, M), \; f$ can be approximated in neighborhoods of compacts by holomorphic mappings whose $k$-jet extensions are transversal to $A$. As an application the stability of Eisenman-Kobayshi intrinsic $k$-measures with respect to deleting analytic subsets of codimension $>k$ is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.


References [Enhancements On Off] (What's this?)

  • AVG V.I. Arnold, A.N. Varchenko, S.M. Gusein-Zade, Singularities of differentiable maps, Boston, Birkhäuser, 1985. MR 86f:58018
  • BG R. Brody, M. Green, On a family of hyperbolic surfaces in $\Bbb P^{3}$, Duke Math. J., 44(1977), no.4, 873-874. MR 56:12331
  • CHO L. Campbell, A. Howard, T. Ochiai, Moving discs off analytic subsets, Proc. AMS, 60(1976), no.1, 106-107. MR 54:13143
  • CO L. Campbell, R. Ogawa, On preserving the Kobayashi pseudometric, Nagoya Math. J., 57(1975), no.1, 37-47. MR 51:8474
  • Ch E.M. Chirka, Complex Analytic Sets, Kluwer Academic Publish., Dordrecht, 1989. MR 92b:32016
  • DLS J.-P. Demailly, L. Lempert, B. Shiffman, Algebraic approximation of holomorphic maps from Stein domains to projective manifolds, preprint, 1993.
  • E D.A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Mem. AMS, no.96, AMS, Providence, R.I., 1970. MR 41:3807
  • G M.L. Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math., 97(1975), 43-75. MR 51:3544
  • GW I. Graham, H. Wu, Some remarks on the intrinsic measures of Eisenman, Trans. A.M.S., 288(1985), 625-660. MR 86e:32031
  • Ka 1 Sh. Kaliman, Some facts about Eisenman intrinsic measures, Complex Variables, (to appear).
  • Ka 2 Sh. Kaliman, Exotic analytic structures and Eisenman intrinsic measures, Israel J. Math, 88 (1994), 411--423. CMP95:04.
  • Ko 1 Sh. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Appl. Math., 2, Dekker, New York, 1970. MR 43:3503
  • Ko 2 Sh. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. AMS, 82(1976), 357-416. MR 54:3032
  • LZ V. Ya. Lin, M.G. Zaidenberg, Finiteness theorems for holomorphic mappings, Encyclopaedia of Math. Sci., 9(1986), 127-194 (in Russian). English transl. in Encyclopaedia of Math. Sci., Vol.9. Several Complex Variables III. Berlin-Heidelberg-New York, Springer Verlag, 1989, 113-172. CMP 19:02
  • P D.A. Pelles ($=$ D.A. Eisenman), Holomorphic maps which preserve intrinsic measure, Amer. J. Math., 97 (1975), 1-15. MR 51:3542
  • PS E.A. Poletsky, B.V. Shabat, Invariant metrics, Encyclopaedia of Math. Sci., 9(1986), 73-125 (in Russian). English transl. in Encyclopaedia of Math. Sci., Vol.9. Several Complex Variables III. Berlin-Heidelberg-New York, Springer Verlag, 1989, 63-111. CMP 19:02
  • Ra V.V. Rabotin, A counterexample to two problems of Kobayashi, Multidimensional Complex Analysis (Russian), 256-257, Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1985. MR 88i:32034
  • Ro H.L. Royden, Remarks on the Kobayashi metric. In Several Complex Variables, II (Maryland, 1970), Lect. Notes Math., 185(1971), 125-137. MR 46:3826
  • S Y.-T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math., 38(1976), 89-100. MR 55:8407
  • T R. Thom, Un lemme sur les applications differentiables, Bol. Soc. Mat. Mexicana (2), 1(1956), 59-71. MR 21:910
  • W H. Whitney, Complex Analytic Varieties, Addison-Wesley, 1972. MR 52:8473

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32E10, 32H02, 58C10, 58A35, 58A07

Retrieve articles in all journals with MSC (1991): 32E10, 32H02, 58C10, 58A35, 58A07


Additional Information

Sh. Kaliman
Affiliation: Department of Mathematics & Computer Science, University of Miami, Coral Gables, Florida 33124
Email: kaliman@paris-gw.cs.miami.edu

M. Zaidenberg
Affiliation: Université Grenoble I, Institut Fourier des Mathématiques, B.P. 74, 38402 Saint Martin d’Hères–Cédex, France
Email: zaidenbe@fourier.grenet.fr

DOI: https://doi.org/10.1090/S0002-9947-96-01482-1
Received by editor(s): November 16, 1994
Additional Notes: Supported by General Research Support Award
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society