Polynomially Convex Embeddings of Even-Dimensional Compact Manifolds

TL;DR

This paper extends the understanding of polynomially convex embeddings of even-dimensional compact manifolds into complex Euclidean spaces, showing new bounds and properties even with complex tangencies.

Contribution

It demonstrates that key properties of polynomially convex embeddings hold for lower dimensions than previously known, specifically when n=3k-1.

Findings

Polynomially convex embeddings are generic for n≥3k-1.

Existence of smooth generators for the Banach algebra C(M).

Embeddings with no analytic disks in their hulls.

Abstract

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth generators for the Banach algebra $\mathcal C(M)$, the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when $n=3k-1$, $k>1$, despite the presence of complex tangencies, thus lowering the known bound for the optimal $n$ in these (related but inequivalent) questions.