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

TL;DR

This paper demonstrates that certain compact strongly pseudoconvex real algebraic hypersurfaces in complex two-space cannot be locally holomorphically embedded into fixed higher-dimensional spheres, revealing limitations in embedding problems.

Contribution

It constructs a family of hypersurfaces in ^2 and proves non-embeddability into spheres of any fixed dimension, advancing understanding of holomorphic embedding obstructions.

Findings

Constructed hypersurfaces in ^2 with specific properties.

Proved non-embeddability into fixed spheres for small perturbations.

Established quantitative bounds psilon(N) for embedding obstructions.

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 $\mathbb{C}^N.$