Rigid equivalences of $5$-dimensional $2$-nondegenerate rigid real hypersurfaces $M^5\subset {\mathbb{C}}^3$ of constant Levi rank $1$

Abstract

We study the local equivalence problem for real-analytic (${\mathcal{C}}^{\omega }$) hypersurfaces $M^5\subset {\mathbb{C}}^3$ which, in coordinates $(z_1,z_2,w)\in {\mathbb{C}}^3$ with $w=u+iv$, are rigid: \[ u \,=\, F\big(z_1,z_2,\overline{z}_1,\overline{z}_2\big), \] with $F$ independent of $v$. Specifically, we study the group ${\mathsf{Hol}}_{\mathsf{rigid}}(M)$ of rigid local biholomorphic transformations of the form: \[ \big(z_1,z_2,w\big) \longmapsto \Big( f_1(z_1,z_2), f_2(z_1,z_2), a\,w + g(z_1,z_2) \Big), \] where $a\in \mathbb{R}\backslash \{0\}$ and $\frac{D(f_1,f_2)}{D(z_1,z_2)}\ne 0$, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate $\{e\}$-structure, we find exactly two primary invariants $I_0$ and $V_0$, which we express explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The identical vanishing $0\equiv I_0\left(J^{5}F\right)\equiv V_0\left(J^{5}F\right)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: \[ M_{\sf LC} \colon \ \ \ \ \ u \,=\, \frac{z_1\,\overline{z}_1 +\frac{1}{2}\,z_1^2\overline{z}_2 +\frac{1}{2}\,\overline{z}_1^2z_2}{ 1-z_2\overline{z}_2}. \] We establish that $\dim {\mathsf{Hol}}_{\mathsf{rigid}}(M)\le 7=\dim {\mathsf{Hol}}_{\mathsf{rigid}}\left(M_{\mathsf{LC}}\right)$ always. If one of these two primary invariants $I_0\not\equiv 0$ or $V_0\not\equiv 0$ does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain $5$-dimensional $\{e\}$-structure on $M$.

This work was supported in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583. The first author is supported by NSFC grant number 11688101. 32 pages