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.Comment: 21 pages, accepted to Arkiv for Mathemati
We study the singular set of a singular Levi-flat real-analytic hypersurface.
We prove that the singular set of such a hypersurface is Levi-flat in the
appropriate sense. We also show that if the singular set is small enough, then
the Levi-foliation extends to a singular codimension one holomorphic foliation
of a neighborhood of the hypersurface.Comment: 21 pages, minor typos fixed, updated references, accepted to Math.
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