CR singularities of real fourfolds in $\mathbb{C}^3$

Coffman, Adam

doi:10.1215/ijm/1286212924

Cited by 26 publications

(37 citation statements)

References 30 publications

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“…If they are real analytic and elliptic (γ < 1), they are also flat (see [24]). In dimension n = 2 a simple description of ∼-congruence classes can be obtained (Coffman [6] and Izotov [21]). In a later section we generalize this result to dimensions 3 and 4 in the case of quadratically flat complex points.…”

Section: Normal Forms Up To Quadratic Termsmentioning

confidence: 99%

“…It reduces the number of parameters by roughly one half. We also note here that the method, which was used in [6] for the case of 2 × 2 matrices (i.e. putting A into a nice form first, and then T -conjugating B by the matrices preserving A under * -conjugation), does not seem to adapt to the case of 3×3 matrices due to the involved computations.…”

Section: For Few Small M's We Can Diagonalize Hermitian Matricesmentioning

confidence: 99%

“…Classifying complex points up to their quadratic term means finding nice normal form representatives for matrices A and B for this congruence relation, and it reduces to a linear algebra problem. The classification is trivial in the case of n = 1, and for n = 2 was done by Coffman [6]. Note that if A is positive definite, we can find a nonsingular P, so that P * AP = I and then use Autonne-Takagi theorem (any complex symmetric matrix is unitary T -congruent to a real diagonal matrix with non-negative entries) to simplify B.…”

Section: Introductionmentioning

confidence: 99%

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On normal forms of complex points of codimension 2 submanifolds

Slapar

Starčič

2018

Journal of Mathematical Analysis and Applications

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Section: Normal Forms Up To Quadratic Termsmentioning

confidence: 99%

Section: For Few Small M's We Can Diagonalize Hermitian Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On normal forms of complex points of codimension 2 submanifolds

Slapar

Starčič

2018

Journal of Mathematical Analysis and Applications

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“…Proposition 6.2. Let M ⊂ C n × R, n ≥ 2, be a smooth submanifold given by (5), Q nondegenerate, a ≥ 2, and write (5) as s = ρ(z,z).…”

Section: Formal Extension At a Cr Singularitymentioning

confidence: 99%

“…Theorem 1.2. Suppose M and H + are defined near the origin by (5) and (6), n ≥ 2, and Q is nondegenerate. Then there exists a neighborhood U of the origin, such that given a smooth CR function f : M → C, we have:…”

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 singularities of real fourfolds in $\mathbb{C}^3$

Cited by 26 publications

References 30 publications

On normal forms of complex points of codimension 2 submanifolds

On normal forms of complex points of codimension 2 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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