The O-minimal Zilber Conjecture in Higher Dimensions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916285837737984 |
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| author | Castle, Benjamin |
| author_facet | Castle, Benjamin |
| contents | We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let $M$ be an interpretable set in $\mathcal R$, and let $\mathcal M=(M,...)$ be a reduct of the induced structure on $M$. If $\mathcal M$ is strongly minimal and not locally modular, then $\dim_{\mathcal R}(M)=2$. As an application, we prove the Zilber trichotomy for all strongly minimal structures interpreted in the theory of compact complex manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The O-minimal Zilber Conjecture in Higher Dimensions Castle, Benjamin Logic Primary 03C45, Secondary 03C64 We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let $M$ be an interpretable set in $\mathcal R$, and let $\mathcal M=(M,...)$ be a reduct of the induced structure on $M$. If $\mathcal M$ is strongly minimal and not locally modular, then $\dim_{\mathcal R}(M)=2$. As an application, we prove the Zilber trichotomy for all strongly minimal structures interpreted in the theory of compact complex manifolds. |
| title | The O-minimal Zilber Conjecture in Higher Dimensions |
| topic | Logic Primary 03C45, Secondary 03C64 |
| url | https://arxiv.org/abs/2406.09285 |