CR singular images of generic submanifolds under holomorphic maps

Jiří Lebl, André Minor, Ravi Shroff, Duong Son, Yuan Zhang

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)301-327
Number of pages27
JournalArkiv for Matematik
Volume52
Issue number2
DOIs
StatePublished - Oct 1 2014

ASJC Scopus subject areas

  • Mathematics(all)

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