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

Abstract

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.

20 pages, accepted version