Open Access
October 2014 CR singular images of generic submanifolds under holomorphic maps
Jiří Lebl, André Minor, Ravi Shroff, Duong Son, Yuan Zhang
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Ark. Mat. 52(2): 301-327 (October 2014). DOI: 10.1007/s11512-013-0193-0


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.

Funding Statement

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.


Download Citation

Jiří Lebl. André Minor. Ravi Shroff. Duong Son. Yuan Zhang. "CR singular images of generic submanifolds under holomorphic maps." Ark. Mat. 52 (2) 301 - 327, October 2014.


Received: 26 November 2012; Published: October 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1310.32040
MathSciNet: MR3255142
Digital Object Identifier: 10.1007/s11512-013-0193-0

Rights: 2014 © Institut Mittag-Leffler

Vol.52 • No. 2 • October 2014
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