The polynomially convex embedding dimension of real manifolds of dimension $\leq 11$

TL;DR

The paper resolves the polynomially convex smooth embedding problem for compact real manifolds in low dimensions by proving that every $n$-manifold with $n\le 11$ embeds into $\mathbb{C}^{n+1}$ as a polynomially convex set, with the image stratified into a totally real part and a totally real CR-singular locus. The authors develop a jet-transversality framework to first make the CR-singular set smooth and then ensure its image is totally real via a careful $J^2$-level perturbation, with codimension constraints guiding the perturbations. For $n\ge 12$, they obtain a weaker bound $N\le \lfloor 5n/4\rfloor-1$ using a similar perturbation strategy, extending the approach and offering a uniform density of holomorphic polynomials on the resulting polynomially convex embeddings. Overall, the work advances the Izzo-Stout program by providing sharp results up to $n=11$ and a substantial bound beyond, with broad implications for approximation theory on polynomially convex sets.

Abstract

We show that any compact smooth real $n$-dimensional manifold $M$ with $n\leq 11$ can be smoothly embedded into $\mathbb{C}^{n+1}$ as a polynomially convex set. In general, there is no such embedding into $\mathbb{C}^n$. This solves a problem by Izzo and Stout for $n\leq 11$. Additionally, we show that the image $\widetilde{M}$ of $M$ in $\mathbb{C}^{n+1}$ is stratified totally real. As a consequence, by a result in [13], each continuous complex-valued functions on $\widetilde{M}$ is the uniform limit on $\widetilde{M}$ of holomorphic polynomials in $\mathbb{C}^{n+1}$. Our proof is based on the jet transversality theorem and a slight improvement of a perturbation result by the first and the third author.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 3.1
• proof : Sketch of proof
• Lemma 3.2
• proof
• Proposition 3.3
• Lemma 4.1