Images of real submanifolds under finite holomorphic mappings

Ebenfelt, Peter; Rothschild, Linda Preiss

doi:10.4310/cag.2007.v15.n3.a2

Cited by 10 publications

(12 citation statements)

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“…If the map F does not have constant rank at a point, the image of that point is a CR singular point of M (see Lemma 4.2). This observation was also made in [7], where the images of CR submanifolds Date: December 2, 2013. The first author was in part supported by NSF grant DMS 0900885.…”

Section: Introductionmentioning

confidence: 61%

CR singular images of generic submanifolds under holomorphic maps

et al. 2014

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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

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Section: Introductionmentioning

confidence: 61%

CR singular images of generic submanifolds under holomorphic maps

et al. 2014

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“…We usually write N = dim C T M = n + d. If (M, V ) and (M , V ) are abstract CR manifolds, and h : M → M is a CR map of class C 1 , we say that h is strictly noncharacteristic if for every p ∈ M , h * (T 0 p M ) = T 0 p M. We should mention that the notion of strictly noncharacteristic map coincides with the well-known condition of CR transversality (see e.g. [10,13]) when M and M have the same CR codimension (see [16]). If h is as above, the singular support of h, denoted SingSupp h, is the locus of points p in M such that h is not C ∞ -smooth in any neighborhood of p.…”

Section: Introduction and Results For Cr Structures Of Hypersurface Typementioning

confidence: 99%

Regularity of CR mappings of abstract CR structures

Lamel

Mir

2019

Int. J. Math.

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We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).

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“…In this one step of the proof of Theorem A, it will be convenient to use Proposition 3.1 to replace X with a singular codimension one holomorphic foliation F that is singular at 0 whose pullback g * F is smooth. Moreover, in the proof of Proposition 3.1 we saw that the resulting foliation satisfies F sing = X sing , so that 0 is an isolated singularity for F. 3 A priori, X could be a reducible hypersurface. We'll rule this out in the last paragraph of the proof.…”

Section: Proof Of Theorems a And Bmentioning

confidence: 96%

“…When X is a germ of an analytic subvariety of C n having arbitrary dimension, a version of Theorem A has been proved by Ebenfelt-Rothschild [3, Theorem 2.1], Lebl [7,Section 4], and Denkowski [2, Theorem 1.2] under the additional hypothesis that the Jacobian det(Dg) does not vanish identically on g −1 (X). However, this hypothesis has been removed in the case that dim(X) = 1, see [3,Theorem 4.1] and [7,Section 4]. We refer the reader to the aforementioned papers for further details.…”

Section: Introductionmentioning

confidence: 99%