CR singular images of generic submanifolds under holomorphic maps

TL;DR

This paper investigates the local geometry of CR singular real-analytic manifolds that are images of CR manifolds under diffeomorphic CR maps, revealing conditions for finite holomorphic extensions, invariants, and properties of Levi-flat and singular images.

Contribution

It establishes a necessary and sufficient condition for CR maps to extend holomorphically, introduces the Moser invariant as a biholomorphic invariant, and analyzes the structure of Levi-flat and singular CR images.

Findings

Finite holomorphic extension condition for CR diffeomorphisms.

Moser invariant as a biholomorphic invariant for Bishop surfaces.

CR functions on singular images may not extend holomorphically.

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.