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 $(\mathbb{P}^{n}, \mathbb{P}^{n-1})$ . 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.
