Polynomially convex embeddings of odd-dimensional closed manifolds

TL;DR

This paper proves that odd-dimensional closed manifolds can be smoothly embedded into complex Euclidean spaces with polynomial convexity, improving previous bounds and enabling approximation by holomorphic polynomials, using a topological approach.

Contribution

It establishes new bounds for polynomially convex embeddings of odd-dimensional manifolds and introduces techniques to handle CR-singularities with topological methods.

Findings

Any smooth closed orientable (2k+1)-manifold embeds into C^{3k} with polynomial convexity.

Embeddings allow uniform approximation of continuous functions by holomorphic polynomials.

Constructs embeddings with hulls containing no nontrivial analytic disks.

Abstract

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible ambient complex dimension for such embeddings (which is sharp when $k=1$). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.