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.

21 pages, accepted to Arkiv for Mathematik

References in corpus (3)

• Singular set of a Levi-flat hypersurface is Levi-flat
• Nowhere minimal CR submanifolds and Levi-flat hypersurfaces
• Flattening of CR singular points and analyticity of local hull of holomorphy

Cited by in corpus (7)

• Normal forms for CR singular codimension two Levi-flat submanifolds
• Codimension two CR singular submanifolds and extensions of CR functions
• Extension of CR functions from boundaries in ${\mathbb{C}}^n\times \mathbb{R}$
• On Lewy extension for smooth hypersurfaces in ${\mathbb{C}}^n\times \mathbb{R}$
• A CR singular analogue of Severi's theorem
• Flattening a non-degenerate CR singular point of real codimension two
• On CR singular CR images