Local Biholomorphic Straightening of Real Submanifolds

Freeman, Michael

doi:10.2307/1971099

Cited by 36 publications

(19 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a consequence, we can define A k+1 , k ≥ 0, inductively to be the smallest linear subspace of A 1,0 that contains A k and L X (A k ) for every X ∈ H For the rest of the section we assume that M has constant degeneracy. For manifolds of this type we give an equivalent approach to finite nondegeneracy using Lie brackets of vector fields rather than Lie derivatives, compare also [7].…”

Section: Appendix: Nondegeneracy Conditionsmentioning

confidence: 99%

“…M is the simplest known real hypersurface in C 3 with everywhere degenerate Levi form that cannot be even locally biholomorphically straightened, i.e. that is not locally CR-equivalent to a direct product S × C with S any real hypersurface in C 2 , compare [7], [5]. M is homogeneous as CR-manifold since the group of all affine transformations of C 3 fixing M acts transitively on M (and H).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Kaup¹,

Zaitsev²

2006

J. Eur. Math. Soc.

View full text Add to dashboard Cite

Section: Appendix: Nondegeneracy Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Kaup¹,

Zaitsev²

2006

J. Eur. Math. Soc.

View full text Add to dashboard Cite

“…REMARK. Freeman [10] has an alternative list of obstructions to straightening. It would be interesting to determine the precise relationships.…”

Section: Lemma Suppose That the Defining Function (H E) Does Not Exmentioning

confidence: 99%

Defining equations for real analytic real hypersurfaces in ${\bf C}\sp n$

D’Angelo¹

1986

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

A defining function for a real analytic real hypersurface can be uniquely written as 2 Ke(H) + E, where H is holomorphic and E contains no pure terms. We study how H and E change when we perform a local biholomorphic change of coordinates, or multiply by a unit. One of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of E (expanded in terms of H) beyond £?oo that serve as obstructions to writing a defining equation as 2 Re(fe) + e, where e is independent of h. We also find necessary pluriharmonic obstructions to doing this, which arise from the easier case of attempting to straighten the hypersurface.

show abstract

“…The classification and equivalence problems of Levi degenerate hypersurfaces in complex spaces are much less understood than the non-degenerate case. The task of identifying suitable higher non-degeneracy conditions was first considered by Freeman [11], and the modern language speaks of '2-nondegeneracy'. An elementary self-contained presentation of foundational aspects is available in [28], and will be enough for our purposes in this paper.…”

Section: Introductionmentioning

confidence: 99%

Rigid Biholomorphic Equivalences of Rigid C2,1 Hypersurfaces M5⊂C3

Foo¹,

Merker²,

Ta³

2023

Michigan Math. J.

View full text Add to dashboard Cite

We study the local equivalence problem for real-analytic (C ω ) hypersurfaces M 5 ⊂ C 3 which, in some holomorphic coordinates (z1, z2, w) ∈ C 3 with w = u + √ −1v, are rigid in the sense that their graphing functions:are independent of v. Specifically, we study the group Hol rigid (M ) of rigid local biholomorphic transformations of the form:where a ∈ R\{0} andwhich preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate {e}-structure, we find exactly two primary invariants I0 and V0, which we express explicitly in terms of the 5-jet of the graphing function F of M . The identical vanishing 0 ≡ I0(J 5 F ) ≡ V0(J 5 F ) then provides a necessary and sufficient condition for M to be locally rigidly-biholomorphic to the known model hypersurface:We establish that dim Hol rigid (M ) 7 = dim Hol rigid (M LC ) always. If one of these two primary invariants I0 ≡ 0 or V0 ≡ 0 does not vanish identically, then on either of the two Zariskiopen sets {p ∈ M : I0(p) = 0} or {p ∈ M : V0(p) = 0}, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain 5-dimensional {e}-structure on M , that is, we get an invariant absolute parallelism on M 5 . Hence dim Hol rigid (M ) drops from 7 to 5, illustrating the gap phenomenon.

show abstract

Local Biholomorphic Straightening of Real Submanifolds

Cited by 36 publications

References 4 publications

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Defining equations for real analytic real hypersurfaces in ${\bf C}\sp n$

Rigid Biholomorphic Equivalences of Rigid C2,1 Hypersurfaces M5⊂C3

Contact Info

Product

Resources

About