New examples of 2-nondegenerate real hypersurfaces in \(\mathbb{C}^N\) with arbitrary nilpotent symbols
type: Item
instance of: scholarly article
title: New examples of 2-nondegenerate real hypersurfaces in \(\mathbb{C}^N\) with arbitrary nilpotent symbols
author: Martin Kolar
author: Ilya Kossovskiy
author: D. Sykes
published in: Journal of the London Mathematical Society. Second Series
publication date: 2024-08-22T00:00:00Z
review text: Let \(\phi (x_1 \, , \, \cdots \, , \, x_n )\) be a \(C^2\) real valued function of \(n\) real variables, and let \N\[\Phi \big( z_1 , \cdots,z_n \big) := \phi \Big( \frac{z_1 + \overline{z}_1}{2} , \cdots , \frac{z_n + \overline{z}_n}{2} \Big)\] with \(z_j = x_j + i \, y_j \in {\mathbb C}\), \(1 \leq j \leq n\), \(i = \sqrt{-1}\). Let \(M_\phi\) be the real hypersurface in \({\mathbb C}^{n+1}\) given by \N\[M_\phi = \big\{ (z_0 \, , \, z_1 \, , \, \cdots \, , \, z_n ) \in {\mathbb C}^{n+1} \; : \; z_0 + \overline{z}_0 = \Phi (z_1 \, , \, \cdots \, , \, z_n ) \big\}.\] \NLet \(T_{1,0}(M_\phi )\) be the CR structure on \(M_\phi\) induced by the complex structure of the ambient space \({\mathbb C}^{n+1}\) and let \(G_\theta\) be the Levi form (associated to a choice of pseudohermitian structure \(\theta\) on \((M_\phi \, , \, T_{1,0}(M_\phi ) )\), cf. [the reviewer and \textit{G. Tomassini} Differential Geometry and Analysis on CR manifolds. Birkhäuser: Boston-Basel-Berlin (2006; Zbl 1099.32008)]. For every \(x \in M_\phi\) let us set \(K_x = \big\{ v \in T_{1,0}(M_\phi )_x \, : \, G_\theta (v, \, \overline{w} ) = 0, \;\; w \in T_{1,0}(M_\phi )_x \big\}\) and assume that \(K \to M_\phi\) is a complex line bundle. Let \(x \in M_\phi\) and \(v \in K_x\) and let us consider the map \(\mathrm{ad}_v : T_{1,0}(M_\phi )_x \big/ K_x \to T_{1,0}(M_\phi )_x \big/ K_x\) given by \(\mathrm{ad}_v \big( W_x + K_x \big) := \pi_x \, \big[ Z, \, \overline{W} \big]_x\) where \(Z\) is a \(C^1\) section in \(K\) extending \(v\), i.e., \(Z_x = v\), and \N\[\pi : T(M_\phi ) \otimes {\mathbb C} \to \big[ T(M_\phi ) \oplus {\mathbb C} \big] \big/ \big[ T_{0,1} (M_\phi ) \oplus K \big]\] is the canonical projection. The authors of the paper under review build new examples of \(2\)-nondegenerate real hypersurfaces \(M_\phi \subset {\mathbb C}^{n+1}\), that is real hypersurfaces \(M_\phi\) such that \(K \to M\) is a complex line bundle and the map \(v \in K_x \longmapsto\mathrm{ad}_v\) is injec…
reviewed by: Sorin Dragomir
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 7900384
zbMATH Keywords: CR structure
zbMATH Keywords: Levi form
zbMATH Keywords: 2-nondegenerate real hypersurface
MaRDI profile type: Publication
cites work: Q4230686
cites work: Regularity of CR mappings between algebraic hypersurfaces
cites work: Model \(CR\) surfaces: weighted approach
cites work: Normal forms and biholomorphic equivalence of real hypersurfaces in C^3
cites work: Uniformly Levi degenerate CR manifolds: The 5-dimensional case
cites work: CR-manifolds of dimension 5: A Lie algebra approach
cites work: Classification of Levi degenerate homogeneous CR-manifolds in dimension 5
cites work: Local tube realizations of CR-manifolds and maximal abelian subalgebras
cites work: Homogeneous submanifolds in the four-dimensional affine and projective geometry
cites work: Local complex foliation of real submanifolds
cites work: Local biholomorphic straightening of real submanifolds
cites work: Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute parallelisms
cites work: On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces
cites work: A complete normal form for everywhere Levi-degenerate hypersurfaces in \(\mathbb{C}^3\)
cites work: On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces
cites work: Two decades of finite jet determination of CR mappings
cites work: Higher order Levi forms on homogeneous CR manifolds
cites work: The equivalence problem for five-dimensional Levi degenerate CR manifolds
cites work: Explicit absolute parallelism for 2-nondegenerate real hypersurfaces \(M^5 \subset \mathbb{C}^3\) of constant Levi rank 1
cites work: Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
cites work: Homogeneous models for Levi degenerate CR manifolds
cites work: Maximal dimension of groups of symmetries of homogeneous 2-nondegenerate CR structures of hypersurface type with a 1-dimensional Levi kernel
cites work: On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations
cites work: On different notions of homogeneity for CR-manifolds
DOI: 10.1112/JLMS.12962
description: scientific article; zbMATH DE number 7900384