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

Abstract: 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 $M_{\epsilon}$ in $\mathbb{C}^2,$ and prove that for any integer $N$, there is a number $\epsilon(N)$ with $0<\epsilon(N)<1$ such that for any $\epsilon$ with $0<\epsilon<\epsilon(N)$, $M_\epsilon$ can not be locally holomorphically embedded into the unit sphere $\mathbb{S}^{2N-1}$ in $\mathbb{C}^N.$