Polynomially convex embeddings of even-dimensional compact manifolds

Abstract

The totally-real embeddability of any 2k-dimensional compact manifold M into C n , n ≥ 3k, has several consequences: the genericity of polynomially convex embeddings of M into C n , the existence of n smooth generators for the Banach algebra 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.pages 897-915, 1995.

Polynomially convex embeddings of odd-dimensional closed manifolds

“…We first isolate a lemma that is used in both the proofs in this section. The proof uses standard arguments; see [18,Lemma 2.1] for a version of this lemma. Here, for p 2 C n and r > 0, B p .r/ D ¹z 2 C n W kz pk < rº and B p .r/ denotes its closure in C n Lemma 5.1.…”

“…In our earlier paper [18], this bound was improved by one for even-dimensional manifolds. That is, for every k 2, .2k/-dimensional manifolds admit polynomially convex smooth embeddings into C 3k 1 .…”

“…For smooth embeddings, the best known bound for surfaces is n 0 Ä 3, obtained in [23] via explicit embeddings. For higher dimensions, n 0 Ä b 3m 2 c 1 when m is even, as seen in [18], and n 0 Ä b 3m 2 c when m is odd, as proved in [5]. For orientable manifolds, we improve the latter as follows.…”

“…We note that totally real embeddability would give an easier proof of (2) and a stronger version of (3) in Theorem 1.1. Finally, in the case of isolated CR-singularities, any embedding of a compact manifold with boundary can be perturbed to be totally real and polynomially convex (see [18]). It is not clear whether our proof can be modified to obtain totally real polynomially convex embeddings of .2k C 1/-dimensional manifolds with boundary in C 3k .…”