Non-embeddability into a fixed sphere for a family of compact real algebraic hypersurfaces

We study the holomorphic embedding problem from a compact strongly pseudoconvex real algebraic hypersurface into a sphere of higher dimension. We construct a family of compact strongly pseudoconvex hypersurfaces $M_{ϵ}$ in ${\mathbb{C}}^2,$ and prove that for any integer $N$, there is a number $ϵ(N)$ with $0<ϵ(N)<1$ such that for any $ϵ$ with $0<ϵ<ϵ(N)$, $M_{ϵ}$ can not be locally holomorphically embedded into the unit sphere ${\mathbb{S}}^{2N-1}$ in ${\mathbb{C}}^N\mathrm{.}$