Wei-Guo Foo, Joël Merker, The-Anh Ta

Michigan Math. J. Advance Publication, 1-26, (2021) DOI: 10.1307/mmj/20205950
KEYWORDS: 58A15, 53A55, 53B25, 53C10, 32V25, 32V40

We study the local equivalence problem for real-analytic (${\mathcal{C}}^{\mathit{\omega}}$) hypersurfaces ${\mathit{M}}^{5}\subset {\mathbb{C}}^{3}$ that, in some holomorphic coordinates $({\mathit{z}}_{1},{\mathit{z}}_{2},\mathit{w})\in {\mathbb{C}}^{3}$ with $\mathit{w}=\mathit{u}+\sqrt{-1}\mathit{v}$, are *rigid* in the sense that their graphing functions

$$\mathit{u}=\mathit{F}({\mathit{z}}_{1},{\mathit{z}}_{2},{\stackrel{\u203e}{\mathit{z}}}_{1},{\stackrel{\u203e}{\mathit{z}}}_{2})$$

are independent of *v*. Specifically, we study the group ${\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})$ of *rigid* local biholomorphic transformations of the form

$$({\mathit{z}}_{1},{\mathit{z}}_{2},\mathit{w})\u27fc({\mathit{f}}_{1}({\mathit{z}}_{1},{\mathit{z}}_{2}),{\mathit{f}}_{2}({\mathit{z}}_{1},{\mathit{z}}_{2}),\mathit{a}\mathit{w}+\mathit{g}({\mathit{z}}_{1},{\mathit{z}}_{2})),$$

where $\mathit{a}\in \mathbb{R}\setminus \{0\}$ and $\mathit{D}({\mathit{f}}_{1},{\mathit{f}}_{2})/\mathit{D}({\mathit{z}}_{1},{\mathit{z}}_{2})\ne 0$, which preserve the rigidity of hypersurfaces.

After performing a Cartan-type reduction to an appropriate $\{\mathit{e}\}$-structure, we find exactly *two* primary invariants ${\mathit{I}}_{0}$ and ${\mathit{V}}_{0}$, which we express explicitly in terms of the 5-jet of the graphing function *F* of *M*. The identical vanishing $0\equiv {\mathit{I}}_{0}({\mathit{J}}^{5}\mathit{F})\equiv {\mathit{V}}_{0}({\mathit{J}}^{5}\mathit{F})$ then provides a necessary and sufficient condition for *M* to be locally *rigidly biholomorphic* to the known model hypersurface

$${\mathit{M}}_{\mathsf{LC}}:\phantom{\rule{1em}{0ex}}\mathit{u}=\frac{{\mathit{z}}_{1}{\stackrel{\u203e}{\mathit{z}}}_{1}+1/2{\mathit{z}}_{1}^{2}{\stackrel{\u203e}{\mathit{z}}}_{2}+1/2{\stackrel{\u203e}{\mathit{z}}}_{1}^{2}{\mathit{z}}_{2}}{1-{\mathit{z}}_{2}{\stackrel{\u203e}{\mathit{z}}}_{2}}.$$

We establish that always $\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})\le 7=\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}({\mathit{M}}_{\mathsf{LC}})$.

If one of these two primary invariants ${\mathit{I}}_{0}\not\equiv 0$ or ${\mathit{V}}_{0}\not\equiv 0$ does not vanish identically, then on either of the two Zariski-open sets $\{\mathit{p}\in \mathit{M}:{\mathit{I}}_{0}(\mathit{p})\ne 0\}$ or $\{\mathit{p}\in \mathit{M}:{\mathit{V}}_{0}(\mathit{p})\ne 0\}$, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain five-dimensional $\{\mathit{e}\}$-structure on *M*, that is, we get an invariant absolute parallelism on ${\mathit{M}}^{5}$. Hence $\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})$ drops from 7 to 5, illustrating the *gap phenomenon.*