Polynomially convex embeddings and CR singularities of real manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916672133136384 |
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| author | Gupta, Purvi Shafikov, Rasul |
| author_facet | Gupta, Purvi Shafikov, Rasul |
| contents | It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal M$ into $\mathbb C^n$ in the $\mathcal C^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal M$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in $\mathbb C^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomially convex embeddings and CR singularities of real manifolds Gupta, Purvi Shafikov, Rasul Complex Variables 32V40, 32E20 It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal M$ into $\mathbb C^n$ in the $\mathcal C^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal M$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in $\mathbb C^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding. |
| title | Polynomially convex embeddings and CR singularities of real manifolds |
| topic | Complex Variables 32V40, 32E20 |
| url | https://arxiv.org/abs/2504.01895 |