CR singular images of generic submanifolds under holomorphic maps

Lebl, Jiri; Minor, André; Shroff, Ravi; Son, Duong Ngoc; Zhang, Yuan

doi:10.1007/s11512-013-0193-0

Cited by 11 publications

(21 citation statements)

References 18 publications

(24 reference statements)

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“…As a corollary of this theorem we obtain in § 8 using the results of [29] that the CR singular set of any type C.1 submanifold is a Levi-flat submanifold of dimension 2n − 2 and CR dimension n − 2.…”

Section: Introductionmentioning

confidence: 88%

“…Let M ⊂ C n+1 be a codimension two Levi-flat CR singular submanifold that is an image of R 2 × C n−1 via a real-analytic CR map, and let S ⊂ M be the CR singular set of M. In [29] it was proved that near a generic point of S exactly one of the following is true:…”

Section: Cr Singular Set Of Type CX Submanifoldsmentioning

confidence: 99%

“…We only have the above classification for a generic point of S, and S need not be a CR submanifold everywhere. See [29] for examples. If M is a Levi-flat CR singular submanifold and the Levi-foliation of M CR extends to M, then by Lemma 6.3 at a generic point S has to be of one of the above types.…”

Section: Cr Singular Set Of Type CX Submanifoldsmentioning

confidence: 99%

“…Real submanifolds with complex tangents have been studied in other situations. See for example [29], where CR singular submanifolds that are images of CR manifolds were studied. Normal forms for the quadratic part of general codimension two CR singular submanifolds in C 3 was completely solved by Coffman [9].…”

Section: Introductionmentioning

confidence: 99%

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Normal forms for CR singular codimension-two Levi-flat submanifolds

Gong

Lebl

2015

Pacific J. Math.

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Abstract. Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to C m , m > 2, of Bishop surfaces in C 2 . Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of z m =z 1 z 2 +z 2 1 , have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in C 3 in the nondegenerate case that shows infinitely many formal invariants.

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Section: Introductionmentioning

confidence: 88%

Section: Cr Singular Set Of Type CX Submanifoldsmentioning

confidence: 99%

Section: Cr Singular Set Of Type CX Submanifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Normal forms for CR singular codimension-two Levi-flat submanifolds

Gong

Lebl

2015

Pacific J. Math.

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“…If M ⊂ C n+1 is not flattenable, then even in the real-analytic case, an extension of CR functions does not in general exist near CR singularities. See Harris [12] and Lebl-Minor-Shroff-Son-Zhang [21].…”

Section: Introductionmentioning

confidence: 99%

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

Lebl

Noell

Ravisankar

2018

Trans. Amer. Math. Soc.

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We prove an analogue of the Lewy extension theorem for a real dimension 2n smooth submanifold M ⊂ C n × R, n ≥ 2. A theorem of Hill and Taiani implies that if M is CR and the Levi-form has a positive eigenvalue restricted to the leaves of C n × R, then every smooth CR function f extends smoothly as a CR function to one side of M . If the Levi-form has eigenvalues of both signs, then f extends to a neighborhood of M . Our main result concerns CR singular manifolds with a nondegenerate quadratic part Q. A smooth CR f extends to one side if the Hermitian part of Q has at least two positive eigenvalues, and f extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed. M = n−1, but dim T (0,1) p M = n is possible. If dim T (0,1) p M = n for all p, then M is a complex submanifold by the Newlander-Nirenberg theorem, and so it is locally equal to C n × {s 0 } for some s 0 , and f extends to both sides of M if and only if f is holomorphic on M. Thus, assume that at least somewhere dim T (0,1) pThe points where dim T (0,1) p M = n − 1 are called CR points of M, and the points where dim T (0,1) p M = n are the so-called CR singularities. Write M CR for the set of CR points of M. We need a definition of a CR function on a possibly CR singular submanifold, and we take the definition in the weakest possible sense: A function f : M → C is CR, if vf = 0 for all CR vector fields on M CR . Alternatively, we obtain the same definition if we allow v to be smooth vector fields on M such that v p ∈ T (0,1) p

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Flattening a non-degenerate CR singular point of real codimension two

Fang

Huang

2018

Geom. Funct. Anal.

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This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in C n+1 with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

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CR singular images of generic submanifolds under holomorphic maps

Cited by 11 publications

References 18 publications

Normal forms for CR singular codimension-two Levi-flat submanifolds

Normal forms for CR singular codimension-two Levi-flat submanifolds

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

Flattening a non-degenerate CR singular point of real codimension two

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