CR singular images of generic submanifolds under holomorphic maps
Abstract
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.
Keywords
Cite
@article{arxiv.1205.5309,
title = {CR singular images of generic submanifolds under holomorphic maps},
author = {Jiri Lebl and André Minor and Ravi Shroff and Duong Son and Yuan Zhang},
journal= {arXiv preprint arXiv:1205.5309},
year = {2015}
}
Comments
21 pages, accepted to Arkiv for Mathematik