The Flat CR Twistor Model $Q^{2,2}$ and Its Algebraic Sections
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910169046188032 |
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| author | Altavilla, Amedeo Marini, Stefano |
| author_facet | Altavilla, Amedeo Marini, Stefano |
| contents | We study the flat CR twistor model $Q^{2,2}\subset \mathbb{CP}^3$ by explicit projective methods. Using the anti-holomorphic involution $j$ associated with the twistor fibration, we classify the projective lines contained in $Q^{2,2}$ into twistor fibres and transverse lines, and relate the latter to round $2$-spheres in $S^3$ through an explicit incidence--tangency correspondence. We classify hyperplane sections under the twistor-compatible symmetry group $PSp(1,1)$ and describe the induced CR geometries on $S^3$. For smooth $j$-invariant quadric sections, we obtain a complete relative classification in terms of Coxeter's inversive distance and show that, in the disjoint case, the construction yields an explicit one-parameter family of globally defined real-analytic non-spherical Levi-nondegenerate CR structures on $S^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24233 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Flat CR Twistor Model $Q^{2,2}$ and Its Algebraic Sections Altavilla, Amedeo Marini, Stefano Differential Geometry Algebraic Geometry Complex Variables 32V05, 32V40, 53C30, 53C28 We study the flat CR twistor model $Q^{2,2}\subset \mathbb{CP}^3$ by explicit projective methods. Using the anti-holomorphic involution $j$ associated with the twistor fibration, we classify the projective lines contained in $Q^{2,2}$ into twistor fibres and transverse lines, and relate the latter to round $2$-spheres in $S^3$ through an explicit incidence--tangency correspondence. We classify hyperplane sections under the twistor-compatible symmetry group $PSp(1,1)$ and describe the induced CR geometries on $S^3$. For smooth $j$-invariant quadric sections, we obtain a complete relative classification in terms of Coxeter's inversive distance and show that, in the disjoint case, the construction yields an explicit one-parameter family of globally defined real-analytic non-spherical Levi-nondegenerate CR structures on $S^3$. |
| title | The Flat CR Twistor Model $Q^{2,2}$ and Its Algebraic Sections |
| topic | Differential Geometry Algebraic Geometry Complex Variables 32V05, 32V40, 53C30, 53C28 |
| url | https://arxiv.org/abs/2604.24233 |