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)



On $CR$-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions

Authors: Ruslan Sharipov and Alexander Sukhov
Journal: Trans. Amer. Math. Soc. 348 (1996), 767-780
MSC (1991): Primary 32D15, 32D99, 32F25, 32H99
MathSciNet review: 1325920
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the algebraicity of smooth $CR$-mappings between algebraic Cauchy-Riemann manifolds. A generalization of separate algebraicity principle is established.

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

  • [Al] H.Alexander, Holomorphic mappings from the ball and polydisk, Math. Ann. 209 (1974), 249--256. MR 50:5018
  • [BM] S.Bochner and W.T.Martin, Several complex variables, Princeton University Press, Princeton, 1948. MR 10:366a
  • [BP] A.Boggess and J.Polking, Holomorphic extension of $CR$ functions, Duke Math. J. 49 (1982), 757--784. MR 84j:32018
  • [Ch] E.Chirka, Introduction to the geometry of $CR$ manifolds, Uspehi Mat. Nauk 46 (1991), 81--164. MR 92m:32012
  • [Fo] F.Forstneric, Mappings of quadric Cauchy - Riemann manifolds, Math. Ann. 292 (1992), 163--180. MR 93e:32033
  • [L] S.Lang, Algebra, Addison--Wesley, Reading, MA, 1965.
  • [Pe] D.Pelles, Proper holomorphic self-maps of the unit ball, Math. Ann. 190 (1971), 298--305. MR 43:3501
  • [Pi] S.I.Pinchuk, Boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki 15 (1974), 205-215. MR 50:2558
  • [Po] H.Poincaré, Les fonctions analytiques de deux variables et la representation conforme, Rend. Circ. Mat. Palermo 23 (1907), 185--220.
  • [RS] M.Reed and B.Simon, Methods of modern mathematical physics. V 1. Functional analysis, Academic Press, New York and London, 1972. MR 58:12429a
  • [Ru] W.Rudin, Function Theory on the Unit Ball of $\mathbb{C}^{\,n}$, Springer, New York, 1980. MR 82i:32002
  • [Su1] A.Sukhov, On $CR$ mappings of real quadric manifolds, Michigan Math. J. 41 (1994), 143--150. MR 94:08
  • [Su2] ------, On holomorphic mappings of domains of the type of ``wedge", Mat. Zametki 52 (1992), 141--145. MR 94e:32047
  • [Su3] ------, On the mapping problem for quadric Cauchy - Riemann manifolds, Indiana Univ. Math. J. 42 (1993), 27--36. MR 94e:32039
  • [Su4] ------, On algebraicity of complex analytic sets, Math. USSR Sbornik 74 (1991), 419--426. MR 93e:32014
  • [Ta] N.Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan 14 (1962), 397--429. MR 26:3086
  • [TH] A.Tumanov and G.Henkin, Local characterization of holomorphic automorphisms of Siegel domains, Funct. Anal. and Appl. 17 (1983), 49--61.
  • [Tu] A.Tumanov, Finite dimensionality of the group of $CR$ automorphisms of standard $CR$ manifolds and proper holomorphic mappings of Siegel domains, Izv. Akad. Nauk USSR Ser. Mat. 52 (1988), 651--659. MR 89f:32061
  • [VW] B.L.Van der Waerden, Algebra 1, Springer Verlag, Berlin--Heidelberg--New York, 1966. MR 41:8186
  • [W1] S.Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53--68. MR 57:3431
  • [W2] ------, Holomorphic mappings of domains with generic corners, Proc. Amer. Math. Soc. 86 (1982), 236--240. MR 83k:32041

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32D15, 32D99, 32F25, 32H99

Retrieve articles in all journals with MSC (1991): 32D15, 32D99, 32F25, 32H99

Additional Information

Ruslan Sharipov
Affiliation: Department of Mathematics, Bashkir State University, Frunze street 32, 450074 Ufa, Russia

Alexander Sukhov
Affiliation: CMI, Universite de Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13, France

Keywords: $CR$-mapping, reflection principle, algebraic $CR$-manifold, separate algebraicity
Received by editor(s): January 26, 1995
Additional Notes: Paper written under the financial support of the International Scientific Foundation of Soros, project #RK4000
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society