Local Polynomial Convexity at Hyperbolic CR-singularities in $M \subset \mathbb{C}^n$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918216105721856 |
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| author | Alagandala, Harshith |
| author_facet | Alagandala, Harshith |
| contents | Let $M$ be a smooth manifold of dimension $n$ embedded in $\mathbb{C}^n$. If $T_pM \subset T_p\mathbb{C}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$. For a generic embedding $M$, we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_pM$ is not totally real. An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case $n=2$, $M^2 \subset \mathbb{C}^2$ is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of $M^n \subset \mathbb{C}^n$ at hyperbolic points in higher dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Polynomial Convexity at Hyperbolic CR-singularities in $M \subset \mathbb{C}^n$ Alagandala, Harshith Complex Variables 32E20 Let $M$ be a smooth manifold of dimension $n$ embedded in $\mathbb{C}^n$. If $T_pM \subset T_p\mathbb{C}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$. For a generic embedding $M$, we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_pM$ is not totally real. An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic. It is known that elliptic points give obstruction to polynomial convexity. In the case $n=2$, $M^2 \subset \mathbb{C}^2$ is locally polynomially convex at a hyperbolic complex point. We investigate local polynomial convexity of $M^n \subset \mathbb{C}^n$ at hyperbolic points in higher dimension. |
| title | Local Polynomial Convexity at Hyperbolic CR-singularities in $M \subset \mathbb{C}^n$ |
| topic | Complex Variables 32E20 |
| url | https://arxiv.org/abs/2511.18190 |