Convergence and finite determination of formal CR mappings

Authors:
M. S. Baouendi, P. Ebenfelt and Linda Preiss Rothschild

Journal:
J. Amer. Math. Soc. **13** (2000), 697-723

MSC (2000):
Primary 32H02

Published electronically:
June 22, 2000

MathSciNet review:
1775734

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

**[A]**M. Artin,*On the solutions of analytic equations*, Invent. Math.**5**(1968), 277–291. MR**0232018****[AM]**M. F. Atiyah and I. G. Macdonald,*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802****[BER1]**M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild,*Algebraicity of holomorphic mappings between real algebraic sets in 𝐶ⁿ*, Acta Math.**177**(1996), no. 2, 225–273. MR**1440933**, 10.1007/BF02392622**[BER2]**M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild,*CR automorphisms of real analytic manifolds in complex space*, Comm. Anal. Geom.**6**(1998), no. 2, 291–315. MR**1651418**, 10.4310/CAG.1998.v6.n2.a3**[BER3]**M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild,*Parametrization of local biholomorphisms of real analytic hypersurfaces*, Asian J. Math.**1**(1997), no. 1, 1–16. MR**1480988**, 10.4310/AJM.1997.v1.n1.a1**[BER4]**M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild,*Real submanifolds in complex space and their mappings*, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR**1668103****[BER5]**-,*Rational dependence of smooth and analytic CR mappings on their jets*, Math. Ann.**315**(1999), 205-249. CMP**2000:04****[BR]**M. S. Baouendi and Linda Preiss Rothschild,*Geometric properties of mappings between hypersurfaces in complex space*, J. Differential Geom.**31**(1990), no. 2, 473–499. MR**1037411****[B]**V. K. Beloshapka,*A uniqueness theorem for automorphisms of a nondegenerate surface in a complex space*, Mat. Zametki**47**(1990), no. 3, 17–22, 141 (Russian); English transl., Math. Notes**47**(1990), no. 3-4, 239–242. MR**1052063**, 10.1007/BF01138501**[BK]**Egbert Brieskorn and Horst Knörrer,*Plane algebraic curves*, Birkhäuser Verlag, Basel, 1986. Translated from the German by John Stillwell. MR**886476****[C]**E. Cartan,*Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I*, Ann. Math. Pura Appl.**11**(1932), 17-90 (or Oeuvres II, 1231-1304).**[DA]**John P. D’Angelo,*Several complex variables and the geometry of real hypersurfaces*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR**1224231****[CM]**S. S. Chern and J. K. Moser,*Real hypersurfaces in complex manifolds*, Acta Math.**133**(1974), 219–271. MR**0425155****[Go1]**Xiang Hong Gong,*Divergence for the normalization of real analytic glancing hypersurfaces*, Comm. Partial Differential Equations**19**(1994), no. 3-4, 643–654. MR**1265811**, 10.1080/03605309408821028**[Go2]**Xianghong Gong,*Divergence of the normalization for real Lagrangian surfaces near complex tangents*, Pacific J. Math.**176**(1996), no. 2, 311–324. MR**1434993****[Gu]**Robert C. Gunning,*Introduction to holomorphic functions of several variables. Vol. I*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR**1052649**

Robert C. Gunning,*Introduction to holomorphic functions of several variables. Vol. II*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990. Local theory. MR**1057177**

Robert C. Gunning,*Introduction to holomorphic functions of several variables. Vol. III*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990. Homological theory. MR**1059457****[Ha]**Robin Hartshorne (ed.),*Algebraic geometry, Arcata 1974*, Proceedings of Symposia in Pure Mathematics, XXIX, American Mathematical Society, Providence, R.I., 1979. Corrected reprint of the 1975 original. MR**600866****[Hu]**Xiaojun Huang,*Schwarz reflection principle in complex spaces of dimension two*, Comm. Partial Differential Equations**21**(1996), no. 11-12, 1781–1828. MR**1421212**, 10.1080/03605309608821246**[Mel]**R. B. Melrose,*Equivalence of glancing hypersurfaces*, Invent. Math.**37**(1976), no. 3, 165–191. MR**0436225****[Mi]**Pierre D. Milman,*Complex analytic and formal solutions of real analytic equations in 𝐶ⁿ*, Math. Ann.**233**(1978), no. 1, 1–7. MR**0486602****[MW]**Jürgen K. Moser and Sidney M. Webster,*Normal forms for real surfaces in 𝐶² near complex tangents and hyperbolic surface transformations*, Acta Math.**150**(1983), no. 3-4, 255–296. MR**709143**, 10.1007/BF02392973**[O]**Toshio Ōshima,*On analytic equivalence of glancing hypersurfaces*, Sci. Papers College Gen Ed. Univ. Tokyo**28**(1978), no. 1, 51–57. MR**0494335****[T]**Noboru Tanaka,*On the pseudo-conformal geometry of hypersurfaces of the space of 𝑛 complex variables*, J. Math. Soc. Japan**14**(1962), 397–429. MR**0145555****[TH]**A. E. Tumanov and G. M. Khenkin,*Local characterization of holomorphic automorphisms of Siegel domains*, Funktsional. Anal. i Prilozhen.**17**(1983), no. 4, 49–61 (Russian). MR**725415****[VW]**B. L. van der Waerden,*Modern Algebra. Vol. I*, Frederick Ungar Publishing Co., New York, N. Y., 1949. Translated from the second revised German edition by Fred Blum; With revisions and additions by the author. MR**0029363****[W]**S. M. Webster,*Holomorphic symplectic normalization of a real function*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**19**(1992), no. 1, 69–86. MR**1183758****[Z]**Dmitri Zaitsev,*Germs of local automorphisms of real-analytic CR structures and analytic dependence on 𝑘-jets*, Math. Res. Lett.**4**(1997), no. 6, 823–842. MR**1492123**, 10.4310/MRL.1997.v4.n6.a4**[ZS]**Oscar Zariski and Pierre Samuel,*Commutative algebra, Volume I*, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR**0090581**

Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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Additional Information

**M. S. Baouendi**

Affiliation:
Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 92093-0112

Email:
sbaouendi@ucsd.edu

**P. Ebenfelt**

Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Email:
ebenfelt@math.kth.se

**Linda Preiss Rothschild**

Affiliation:
Department of Mathematics, 0112, University of California at San Diego, La Jolla, California 92093-0112

Email:
lrothschild@ucsd.edu

DOI:
https://doi.org/10.1090/S0894-0347-00-00343-X

Keywords:
Formal mappings,
generic submanifolds,
CR mappings,
holomorphic mappings,
finite determination

Received by editor(s):
June 3, 1999

Published electronically:
June 22, 2000

Additional Notes:
The first and the third authors are partially supported by National Science Foundation grant DMS 98-01258. The second author is partially supported by a grant from the Swedish Natural Science Research Council.

Article copyright:
© Copyright 2000
American Mathematical Society