Arkiv för Matematik
- Ark. Mat.
- Volume 52, Number 2 (2014), 301-327.
CR singular images of generic submanifolds under holomorphic maps
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.
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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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. http://projecteuclid.org/euclid.afm/1485802681.