New examples of 2-nondegenerate real hypersurfaces in \(\mathbb{C}^N\) with arbitrary nilpotent symbols (Q6591569)

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 injective. One such example, as provided in the paper under review, is \(\phi (x_1 \, , \, \cdots \, , \, x_n ) = f(x_1 ) + \sum_{0 < j < j + k \leq n} x_j \, x_k \, x_n^{n - j -k}\) with \(f \in C^2 ({\mathbb R})\).