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

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.