On the Classification of Homogeneous Hypersurfaces in Complex Space

TL;DR

This paper investigates the embeddability of specific homogeneous hypersurfaces in complex spaces, proving non-embeddability in certain cases and establishing conditions under which embeddability occurs, thus advancing the classification of these geometric structures.

Contribution

It demonstrates non-embeddability of $M_t^7$ in ${f C}^7$ and shows embeddability of $M_t^3$ in ${f C}^3$ for a range of $t$, providing new insights into their geometric properties.

Findings

$M_t^7$ is not embeddable in ${f C}^7$ for all $t$

$M_t^3$ is embeddable in ${f C}^3$ for $1<t<1+10^{-6}$

Conjecture: embeddability of $M_t^3$ for all $t$ in $(1, \, ext{approximately } 1.414)$

Abstract

We discuss a family $M_t^n$, with $n\ge 2$, $t>1$, of real hypersurfaces in a complex affine $n$-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in ${\mathbb C}^n$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of $M_t^n$ in ${\mathbb C}^n$ for $n=3,7$. We show that $M_t^7$ is not embeddable in ${\mathbb C}^7$ for every $t$ and that $M_t^3$ is embeddable in ${\mathbb C}^3$ for all $1<t<1+10^{-6}$. As a consequence of our analysis of a map constructed by Ahern and Rudin, we also conjecture that the embeddability of $M_t^3$ takes place for all\, $1<t<\sqrt{(2+\sqrt{2})/3}$.