K\'{a}hler compactification of $\mathbb{C}^n$ and Reeb dynamics

Autor:	Li, Chi, Zhou, Zhengyi
Rok vydání:	2024
Předmět:	Mathematics - Differential Geometry Mathematics - Algebraic Geometry Mathematics - Symplectic Geometry
Druh dokumentu:	Working Paper
Popis:	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, we derive a new characterization of minimal discrepancy of isolated Fano cone singularities by using $S^1$-equivariant positive symplectic homology. Comment: 18 pages, submitted version, added example and a postscript note. Comments very welcome
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2409.10275 Zobrazit plný text záznamu View this record from Arxiv