Models of $2$-nondegenerate CR hypersurface in $\mathbb{C}^N$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910404461985792 |
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| author | Gregorovič, Jan Kolář, Martin Sykes, David |
| author_facet | Gregorovič, Jan Kolář, Martin Sykes, David |
| contents | We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics do in the Levi nondegenerate case. We characterize all $2$-nondegenerate models and show that the moduli space of such hypersurfaces in $\mathbb{C}^N$ is infinite dimensional for each $N>3$. We derive a normal form for these models' defining equations that is unique up to an action of a finite dimensional Lie group. We generalize recently introduced CR invariants termed modified symbols, and show how to compute these intrinsically defined invariants from a model's defining equation. We show that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel and derive explicit formulas for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06525 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Models of $2$-nondegenerate CR hypersurface in $\mathbb{C}^N$ Gregorovič, Jan Kolář, Martin Sykes, David Complex Variables Differential Geometry 32V05, 32V40, 53C30 We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics do in the Levi nondegenerate case. We characterize all $2$-nondegenerate models and show that the moduli space of such hypersurfaces in $\mathbb{C}^N$ is infinite dimensional for each $N>3$. We derive a normal form for these models' defining equations that is unique up to an action of a finite dimensional Lie group. We generalize recently introduced CR invariants termed modified symbols, and show how to compute these intrinsically defined invariants from a model's defining equation. We show that these models automatically possess infinitesimal symmetries spanning a complement to their Levi kernel and derive explicit formulas for them. |
| title | Models of $2$-nondegenerate CR hypersurface in $\mathbb{C}^N$ |
| topic | Complex Variables Differential Geometry 32V05, 32V40, 53C30 |
| url | https://arxiv.org/abs/2404.06525 |