Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916980825522176 |
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| author | Li, Chi Zhou, Zhengyi |
| author_facet | Li, Chi Zhou, Zhengyi |
| contents | Let $X$ be a smooth complex manifold. Assume that $Y\subset X$ is a Kähler submanifold such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example $(\mathbb{P}^n, \mathbb{P}^{n-1})$. We then study certain Kähler orbifold compactifications of $\mathbb{C}^n$ and, as an application, prove that on $\mathbb{C}^3$ the flat metric is the only asymptotically conical Ricci-flat Kähler metric whose metric cone at infinity has a smooth link. As a key technical ingredient, we derive a new characterization of minimal discrepancy of isolated Fano cone singularities by using $S^1$-equivariant positive symplectic homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics Li, Chi Zhou, Zhengyi Differential Geometry Algebraic Geometry Symplectic Geometry Let $X$ be a smooth complex manifold. Assume that $Y\subset X$ is a Kähler submanifold such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example $(\mathbb{P}^n, \mathbb{P}^{n-1})$. We then study certain Kähler orbifold compactifications of $\mathbb{C}^n$ and, as an application, prove that on $\mathbb{C}^3$ the flat metric is the only asymptotically conical Ricci-flat Kähler metric whose metric cone at infinity has a smooth link. As a key technical ingredient, we derive a new characterization of minimal discrepancy of isolated Fano cone singularities by using $S^1$-equivariant positive symplectic homology. |
| title | Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics |
| topic | Differential Geometry Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2409.10275 |