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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2505.12416 |
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| _version_ | 1866918024055881728 |
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| author | Asano, Takumi |
| author_facet | Asano, Takumi |
| contents | Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least $2$ is bijective. We prove Miyanishi conjecture for any quasi-projective variety $X$ which is a dense open subset of a $\mathbb{Q}$-factorial normal projective variety $\overline{X}$ such that codim $(\overline{X} \setminus X) \ge 2$ with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that $\overline{X}$ has canonical singularities and $\overline{X}$ has the canonical model which is obtained by divisorial contractions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12416 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Miyanishi conjecture for quasi-projective varieties Asano, Takumi Algebraic Geometry Dynamical Systems 14A10(Primary), 14E30(Secondary) Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least $2$ is bijective. We prove Miyanishi conjecture for any quasi-projective variety $X$ which is a dense open subset of a $\mathbb{Q}$-factorial normal projective variety $\overline{X}$ such that codim $(\overline{X} \setminus X) \ge 2$ with the ample canonical divisor or the ample anti-canonical divisor. Also, we observe Miyanishi conjecture without the conditions of its canonical divisor by using minimal model program. In particular, we prove Miyanishi conjecture in the case that $\overline{X}$ has canonical singularities and $\overline{X}$ has the canonical model which is obtained by divisorial contractions. |
| title | On Miyanishi conjecture for quasi-projective varieties |
| topic | Algebraic Geometry Dynamical Systems 14A10(Primary), 14E30(Secondary) |
| url | https://arxiv.org/abs/2505.12416 |