Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics

Chi Li, Zhengyi Zhou
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引用次数: 0

Abstract

Let $X$ be a smooth complex manifold. Assume that $Y\subset X$ is a K\"{a}hler 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 K\"{a}hler orbifold compactifications of $\mathbb{C}^n$ and prove that on $\mathbb{C}^3$ the flat metric is the only asymptotically conical Ricci-flat K\"{a}hler metric whose metric cone at infinity has a smooth link. As a key technical ingredient, a new formula for minimal discrepancy of isolated Fano cone singularities in terms of generalized Conley-Zehnder indices in symplectic geometry is derived.
$\mathbb{C}^n$的凯勒紧凑化与里布动力学
让 $X$ 是一个光滑的复流形。假设 $Y/subset X$ 是一个 K\"{a}hler 子流形,使得 $X/setminus Y$ 与 $\mathbb{C}^n$ 是双全向的。我们证明$(X, Y)$ 与标准范例$(\mathbb{P}^n, \mathbb{P}^{n-1})$是双全同的。然后,我们研究了 $\mathbb{C}^n$ 的某些 K\"{a}hler orbifoldcompactifications,并证明在 $\mathbb{C}^3$ 上,平公设是唯一渐近圆锥形的 Ricci-flat K\"{a}hler 公设,它的公设锥在无穷远处有一个光滑链接。作为一个关键的技术成分,以交映几何中的广义康利-泽恩德指数推导出了孤立法诺锥奇点最小差异的新公式。
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