Arkiv för Matematik

  • Ark. Mat.
  • Volume 52, Number 2 (2014), 301-327.

CR singular images of generic submanifolds under holomorphic maps

Jiří Lebl, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang

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The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image, when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.


The first author was in part supported by NSF grant DMS 0900885. The fourth author was in part supported by a scholarship from the Vietnam Education Foundation. The fifth author was in part supported by NSF grant DMS 1265330.

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Ark. Mat. Volume 52, Number 2 (2014), 301-327.

Received: 26 November 2012
First available in Project Euclid: 30 January 2017

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2014 © Institut Mittag-Leffler


Lebl, Jiří; Minor, André; Shroff, Ravi; Son, Duong; Zhang, Yuan. CR singular images of generic submanifolds under holomorphic maps. Ark. Mat. 52 (2014), no. 2, 301--327. doi:10.1007/s11512-013-0193-0.

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  • Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Monodromy and asymptotics of integrals, in Singularities of Differentiable Maps. Vol. II, Monographs in Mathematics 83, Birkhäuser, Boston, MA, 1988.
  • Baouendi, M. S., Ebenfelt, P. and Rothschild, L. P., CR automorphisms of real analytic manifolds in complex space, Comm. Anal. Geom. 6 (1998), 291–315.
  • Baouendi, M. S., Ebenfelt, P. and Rothschild, L. P., Real Submanifolds in Complex Space and Their Mappings, Princeton Mathematical Series 47, Princeton University Press, Princeton, NJ, 1999.
  • Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21.
  • Burns, D. and Gong, X., Singular Levi-flat real analytic hypersurfaces, Amer. J. Math. 121 (1999), 23–53.
  • Diederich, K. and Fornæss, J. E., Pseudoconvex domains with real-analytic boundary, Ann. of Math. 107 (1978), 371–384.
  • Ebenfelt, P. and Rothschild, L. P., Images of real submanifolds under finite holomorphic mappings, Comm. Anal. Geom. 15 (2007), 491–507.
  • Gong, X., Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics, Indiana Univ. Math. J. 53 (2004), 83–95.
  • Harris, G. A., The traces of holomorphic functions on real submanifolds, Trans. Amer. Math. Soc. 242 (1978), 205–223.
  • Harris, G. A., Lowest order invariants for real-analytic surfaces in C2, Trans. Amer. Math. Soc. 288 (1985), 413–422.
  • Huang, X., On an n-manifold in Cn near an elliptic complex tangent, J. Amer. Math. Soc. 11 (1998), 669–692.
  • Huang, X. and Krantz, S. G., On a problem of Moser, Duke Math. J. 78 (1995), 213–228.
  • Huang, X. and Yin, W., A Bishop surface with a vanishing Bishop invariant, Invent. Math. 176 (2009), 461–520.
  • Huang, X. and Yin, W., Flattening of CR singular points and analyticity of local hull of holomorphy. Preprint, 2012.
  • Kenig, C. E. and Webster, S. M., On the hull of holomorphy of an n-manifold in Cn, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (1984), 261–280.
  • Lebl, J., Nowhere minimal CR submanifolds and Levi-flat hypersurfaces, J. Geom. Anal. 17 (2007), 321–341.
  • Lebl, J., Singular set of a Levi-flat hypersurface is Levi-flat, Math. Ann. 355 (2013), 1177–1199.
  • Moser, J., Analytic surfaces in C2 and their local hull of holomorphy, Ann. Acad. Sci. Fenn. Math. 10 (1985), 397–410.
  • Moser, J. K. and Webster, S. M., Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations, Acta Math. 150 (1983), 255–296.
  • Shabat, B. V., Functions of several variables, in Introduction to Complex Analysis. Part II, Translations of Mathematical Monographs 110, Amer. Math. Soc., Providence, RI, 1992.
  • Whitney, H., Complex Analytic Varieties, Addison-Wesley, Reading, MA, 1972.