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CR singular images of generic submanifolds under holomorphic maps

Jiří Lebl Department of Mathematics, University of Wisconsin André Minor Department of Mathematics, University of California at San Diego Ravi Shroff Centre for Mathematics and its Applications Mathematical Sciences Institute, Australian National University Duong Son Department of Mathematics, University of California Yuan Zhang Department of Mathematical Sciences, Indiana University-Purdue University

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1701.10015

Arkiv for Matematik, 52, (2), 301-327, 2012.11
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.
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@inproceedings{jiří2012cr,
  title={CR singular images of generic submanifolds under holomorphic maps},
  author={Jiří Lebl, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203638451807068},
  booktitle={Arkiv for Matematik},
  volume={52},
  number={2},
  pages={301-327},
  year={2012},
}
Jiří Lebl, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang. CR singular images of generic submanifolds under holomorphic maps. 2012. Vol. 52. In Arkiv for Matematik. pp.301-327. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203638451807068.
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