• DOI:10.1515/crelle-2021-0021
• Corpus ID: 221971398

Polynomially convex embeddings of odd-dimensional closed manifolds

@article{Gupta2020PolynomiallyCE,
  title={Polynomially convex embeddings of odd-dimensional closed manifolds},
  author={Purvi Gupta and Rasul Shafikov},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={2020},
  volume={2021},
  pages={273 - 299}
}

• Purvi Gupta, R. Shafikov
• Published 26 September 2020
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)

Abstract It is shown that any smooth closed orientable manifold of dimension 2⁢k+1{2k+1}, k≥2{k\geq 2}, admits a smooth polynomially convex embedding into ℂ3⁢k{\mathbb{C}^{3k}}. This improves by 1 the previously known lower bound of 3⁢k+1{3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k=1{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.