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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2312.02795 |
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| _version_ | 1866909476547723264 |
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| author | Alarcon, Antonio Larusson, Finnur |
| author_facet | Alarcon, Antonio Larusson, Finnur |
| contents | Let $M$ be an open Riemann surface and $A$ be the punctured cone in $\mathbb{C}^n\setminus\{0\}$ on a smooth projective variety $Y$ in $\mathbb{P}^{n-1}$. Recently, Runge approximation theorems with interpolation for holomorphic immersions $M\to\mathbb{C}^n$, directed by $A$, have been proved under the assumption that $A$ is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by $A$ from a smooth affine curve $M$ into $\mathbb{C}^n$. The Oka property is naturally replaced by the stronger assumption that $A$ is algebraically elliptic, which it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02795 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regular immersions directed by algebraically elliptic cones Alarcon, Antonio Larusson, Finnur Complex Variables Algebraic Geometry Differential Geometry Let $M$ be an open Riemann surface and $A$ be the punctured cone in $\mathbb{C}^n\setminus\{0\}$ on a smooth projective variety $Y$ in $\mathbb{P}^{n-1}$. Recently, Runge approximation theorems with interpolation for holomorphic immersions $M\to\mathbb{C}^n$, directed by $A$, have been proved under the assumption that $A$ is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by $A$ from a smooth affine curve $M$ into $\mathbb{C}^n$. The Oka property is naturally replaced by the stronger assumption that $A$ is algebraically elliptic, which it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest. |
| title | Regular immersions directed by algebraically elliptic cones |
| topic | Complex Variables Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2312.02795 |