Acta Mathematica

, 183:273 | Cite as

Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces

  • Dmitri Zaitsev


Complex Space Local Holomorphisms 


  1. [BER1]Baouendi, M. S., Ebenfelt, P. &Rothschild, L. P., Algebraicity of holomorphic mappings between real algebraic sets inC n.Acta Math., 177 (1996), 225–273.MathSciNetMATHGoogle Scholar
  2. [BER2]—, CR automorphisms of real analytic manifolds in complex space.Comm. Anal. Geom., 6 (1998), 291–315.MathSciNetMATHGoogle Scholar
  3. [BER3]—, Parametrization of local biholomorphisms of real analytic hypersurfaces.Asian J. Math., 1 (1997), 1–16.MathSciNetMATHGoogle Scholar
  4. [BER4]—,Real Submanifolds in Complex Space and Their Mappings. Princeton Math. Ser., 47. Princeton Univ. Press, Princeton, NJ, 1999.MATHGoogle Scholar
  5. [BER5]—, Rational dependence of smooth and analytic CR mappings on their jets.Math. Ann., 315 (1999), 205–249.CrossRefMathSciNetMATHGoogle Scholar
  6. [BeRi]Benedetti, R. &Risler, J.-J.,Real Algebraic and Semi-Algebraic Sets. Actualités Mathématiques. Hermann, Paris 1990.MATHGoogle Scholar
  7. [BHR]Baouendi, M. S., Huang, X. &Rothschild, L. P., Regularity of CR mappings between algebraic hypersurfaces.Invent. Math., 125 (1996), 13–36.CrossRefMathSciNetMATHGoogle Scholar
  8. [BJT]Baouendi, M. S., Jacobowitz, H. &Trèves, F., On the analyticity of CR mappings.Ann. of Math., 122 (1985), 365–400.CrossRefMathSciNetGoogle Scholar
  9. [BlGr]Bloom, T. &Graham, I., On type conditions for generic real submanifolds ofC n.Invent. Math., 40 (1977), 217–243.CrossRefMathSciNetGoogle Scholar
  10. [Bo]Boggess, A.,CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1991.MATHGoogle Scholar
  11. [BR1]Baouendi, M. S. &Rothschild, L. P., Mappings of real algebraic hypersurfaces.J. Amer. Math. Soc., 8 (1995), 997–1015.CrossRefMathSciNetMATHGoogle Scholar
  12. [BR2]—, Germs of CR maps between real analytic hypersurfaces.Invent. Math., 93 (1988), 481–500.CrossRefMathSciNetMATHGoogle Scholar
  13. [CMS]Coupet, B., Meylan, F. & Sukhov, A., Holomorphic maps of algebraic CR manifolds.Internat. Math. Res. Notices, 1999, 1–27.Google Scholar
  14. [D]D'Angelo, J. P., Real hypersurfaces, orders of contact, and applications.Ann. of Math., 115 (1982), 615–637.CrossRefMATHMathSciNetGoogle Scholar
  15. [DF1]Diederich, K. &Fornæss, J. E., Pseudoconvex domains with real-analytic boundaries.Ann. of Math., 107 (1978), 371–384.CrossRefGoogle Scholar
  16. [DF2]—, Proper holomorphic mappings between real analytic pseudoconvex domains.Math. Ann., 282 (1988), 681–700.CrossRefMathSciNetMATHGoogle Scholar
  17. [DP1]Diederich, K. &Pinchuk, S., Proper holomorphic maps in dimension 2 extend.Indiana Univ. Math. J., 44 (1995), 1089–1126.CrossRefMathSciNetMATHGoogle Scholar
  18. [DP2]Diederich, K. & Pinchuk, S., Reflection principle in higher dimensions, inProceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math., 1998, Extra Vol. II, pp. 703–712 (electronic).Google Scholar
  19. [DW]Diederich, K. &Webster, S., A reflection principle for degenerate real hypersurfaces.Duke Math. J., 47 (1980), 835–843.CrossRefMathSciNetMATHGoogle Scholar
  20. [Fa]Faran, J., A reflection principle for proper holomorphic mappings and geometric invariants.Math. Z., 203 (1990), 363–377.MATHMathSciNetGoogle Scholar
  21. [Fo1]Forstnerič, F., Extending proper holomorphic mappings of positive codimension.Invent. Math., 95 (1989), 31–62.CrossRefMathSciNetMATHGoogle Scholar
  22. [Fo2]Forstnerič, F., Mappings of quadric Cauchy-Riemann manifolds.Math. Ann., 292 (1992), 163–180.CrossRefMathSciNetMATHGoogle Scholar
  23. [G]Gunning, R. C.,Introduction to Holomorphic Functions of Several Variables (three volumes). Wadsworth & Brooks/Cole Math. Ser. Wadsworth & Brooks/Cole Adv. Books Software, Monterey, CA, 1990.Google Scholar
  24. [H1]Huang, X., On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions.Ann. Inst. Fourier (Grenoble), 44 (1994), 433–463.MATHMathSciNetGoogle Scholar
  25. [H2]—, Geometric analysis in several complex variables. PhD thesis, Washington University, St Louis, MO, 1994.Google Scholar
  26. [H3]—, Schwarz reflection principle in complex spaces of dimension two.Comm. Partial Differential Equations 21 (1996), 1781–1828.MATHMathSciNetGoogle Scholar
  27. [HJ]Huang, X. &Ji, S., Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains.Math. Res. Lett., 5 (1998), 247–260.MathSciNetMATHGoogle Scholar
  28. [HZ]Huckleberry, A. &Zaitsev, D., Actions of groups of birationally extendible automorphisms, inGeometric Complex Analysis (Hayama, 1995), pp. 261–285. World Sci. Publishing, River Edge, NJ, 1996.Google Scholar
  29. [K]Kohn, J. J., Boundary behavior of\(\bar \partial \) on weakly pseudo-convex manifolds of dimension two.J. Differential Geom., 6 (1972), 523–542.MATHMathSciNetGoogle Scholar
  30. [L1]Lewy, H., On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables.Ann. of Math., 64 (1956), 514–522.CrossRefMATHMathSciNetGoogle Scholar
  31. [L2]—, On the boundary behaviour of holomorphic mappings. Contrib. Centro Linceo Inter. Sc. Mat. e Loro Appl., 35, pp. 1–8. Accad. Naz. dei Lincei, Rome, 1977.Google Scholar
  32. [Me]Meylan, F., The reflection principle in complex space.Indiana Univ. Math. J., 44 (1995), 783–796.CrossRefMATHMathSciNetGoogle Scholar
  33. [Mi]Mir, N., Germs of holomorphic mappings between real algebraic hypersurfaces.Ann. Inst. Fourier (Grenoble), 48 (1998), 1025–1043.MATHMathSciNetGoogle Scholar
  34. [Mu]Mumford, D.,The Red Book of Varieties and Schemes. Lecture Notes in Math., 1358. Springer-Verlag, Berlin-New York, 1988.MATHGoogle Scholar
  35. [Pi]Pinchuk, S., On the analytic continuation of holomorphic mappings.Math. USSR-Sb., 27 (1975), 345–392.Google Scholar
  36. [Po]Poincaré, H., Les fonctions analytiques de deux variables et la représentation conforme.Rend. Circ. Mat. Palermo, 23 (1907), 185–220.MATHCrossRefGoogle Scholar
  37. [Se]Segre, B., Intorno al problema di Poincaré della rappresentazione pseudoconforme.Atti R. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (6), 13 (1931), 676–683.MATHMathSciNetGoogle Scholar
  38. [SS]Sharipov, R. &Sukhov, A., On CR-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions.Trans. Amer. Math. Soc., 348 (1996), 767–780.CrossRefMathSciNetMATHGoogle Scholar
  39. [Su1]Sukhov, A., Holomorphic mappings of wedge-type domains.Math. Notes, 52 (1992), 979–982.CrossRefMATHMathSciNetGoogle Scholar
  40. [Su2]—, On algebraicity of complex analytic sets.Math. USSR-Sb., 74 (1993), 419–426.CrossRefMathSciNetGoogle Scholar
  41. [Su3]—, On the mapping problem for quadric Cauchy-Riemann manifolds.Indiana Univ. Math. J., 42 (1993), 27–36.CrossRefMATHMathSciNetGoogle Scholar
  42. [Su4]—, On CR mappings of real quadric manifolds.Michigan Math. J., 41 (1994), 143–150.CrossRefMATHMathSciNetGoogle Scholar
  43. [Ta]Tanaka, N., On the pseudo-conformal geometry of hypersurfaces of the space ofn complex variables.J. Math. Soc. Japan, 14 (1962), 397–429.MATHMathSciNetCrossRefGoogle Scholar
  44. [TH]Tumanov, A. &Henkin, G., Local characterization of holomorphic automorphisms of Siegel domains.Functional Anal. Appl., 17 (1983), 49–61.MathSciNetGoogle Scholar
  45. [Tu1]Tumanov, A., Extension of CR-functions into a wedge from a manifold of finite type.Math. USSR-Sb., 64 (1989), 129–140.CrossRefMATHMathSciNetGoogle Scholar
  46. [Tu2]—, Finite-dimensionality of the group of CR-automorphisms of standard CR-manifolds and characteristic holomorphic mappings of Siegel domains.Math. USSR-Izv., 32 (1989), 655–662.CrossRefMATHMathSciNetGoogle Scholar
  47. [W1]Webster, S., On the mapping problem for algebraic real hypersurfaces.Invent. Math., 43 (1977), 53–68.CrossRefMATHMathSciNetGoogle Scholar
  48. [W2]— On the reflection principle in several complex variables.Proc. Amer. Math. Soc., 71 (1978), 26–28.CrossRefMATHMathSciNetGoogle Scholar
  49. [Z1]Zaitsev, D., On the automorphism groups of algebraic bounded domains.Math. Ann., 302 (1995), 105–129.CrossRefMATHMathSciNetGoogle Scholar
  50. [Z2]—, Germs of local automorphisms of real-analytic CR structures and analytic dependence onk-jets.Math. Res. Lett., 4 (1997), 823–842.MATHMathSciNetGoogle Scholar
  51. [Z3]—, Domains of polyhedral type and boundary extensions of biholomorphisms.Indiana Univ. Math. J., 47 (1998), 1511–1526.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 1999

Authors and Affiliations

  • Dmitri Zaitsev
    • 1
  1. 1.Mathematisches InstitutEberhard-Karls-Universität TübingenTübingenGermany

Personalised recommendations