The moduli space of a rational map is Carathéodory hyperbolic
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913302614900736 |
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| author | Ji, Zhuchao Xie, Junyi |
| author_facet | Ji, Zhuchao Xie, Junyi |
| contents | Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal{M}_f\to X_f$ such that $i(\mathcal{M}_f)$ is precompact in $X_f$. In particular $\mathcal{M}_f$ is Carathéodory hyperbolic (i.e. bounded holomorphic functions separate points in $\mathcal{M}_f$), provided that $f$ is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The moduli space of a rational map is Carathéodory hyperbolic Ji, Zhuchao Xie, Junyi Complex Variables Algebraic Geometry Dynamical Systems Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal{M}_f\to X_f$ such that $i(\mathcal{M}_f)$ is precompact in $X_f$. In particular $\mathcal{M}_f$ is Carathéodory hyperbolic (i.e. bounded holomorphic functions separate points in $\mathcal{M}_f$), provided that $f$ is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism. |
| title | The moduli space of a rational map is Carathéodory hyperbolic |
| topic | Complex Variables Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2404.04568 |