CR singular images of generic submanifolds under holomorphic maps

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

Research output: Contribution to journalArticle

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

Fingerprint

Holomorphic Maps
Submanifolds
Diffeomorphism
Singular Point
Invariant
Analytic function
CR Functions
CR Manifold
Higher Dimensions
Multiplicity
Sufficient
Necessary

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

CR singular images of generic submanifolds under holomorphic maps. / Lebl, Jiří; Minor, André; Shroff, Ravi; Son, Duong; Zhang, Yuan.

In: Arkiv for Matematik, Vol. 52, No. 2, 01.10.2014, p. 301-327.

Research output: Contribution to journalArticle

Lebl, Jiří ; Minor, André ; Shroff, Ravi ; Son, Duong ; Zhang, Yuan. / CR singular images of generic submanifolds under holomorphic maps. In: Arkiv for Matematik. 2014 ; Vol. 52, No. 2. pp. 301-327.
@article{75535bd61164466bba60575843fdeea6,
title = "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.",
author = "Jiř{\'i} Lebl and Andr{\'e} Minor and Ravi Shroff and Duong Son and Yuan Zhang",
year = "2014",
month = "10",
day = "1",
doi = "10.1007/s11512-013-0193-0",
language = "English (US)",
volume = "52",
pages = "301--327",
journal = "Arkiv for Matematik",
issn = "0004-2080",
publisher = "Springer Netherlands",
number = "2",

}

TY - JOUR

T1 - CR singular images of generic submanifolds under holomorphic maps

AU - Lebl, Jiří

AU - Minor, André

AU - Shroff, Ravi

AU - Son, Duong

AU - Zhang, Yuan

PY - 2014/10/1

Y1 - 2014/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84908121890&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908121890&partnerID=8YFLogxK

U2 - 10.1007/s11512-013-0193-0

DO - 10.1007/s11512-013-0193-0

M3 - Article

VL - 52

SP - 301

EP - 327

JO - Arkiv for Matematik

JF - Arkiv for Matematik

SN - 0004-2080

IS - 2

ER -