Kähler compactification of $\mathbb{C}^n$ and Reeb dynamics
Chi Li, Zhengyi Zhou
TL;DR
The paper establishes a bridge between complex-analytic compactifications of $\mathbb{C}^n$ and symplectic invariants of the associated cone singularities. It proves that any pair $(X,Y)$ with $X$ compact and $X\setminus Y\cong \mathbb{C}^n$ is standard under a Kähler assumption, using $S^1$-equivariant positive symplectic homology to compute minimal discrepancies of Fano cone singularities via Reeb dynamics. A new formula $2\mathrm{md}(o,\mathcal{C})=\inf_{\gamma} \mathrm{lSFT}_{\eta}(\gamma)$ ties minimal discrepancy to $\mathrm{lSFT}$ spectra and $SH^{+,S^1}_*$, enabling rigidity results for asymptotically conical (AC) Ricci-flat Kähler metrics; in particular, on $\mathbb{C}^3$ the flat metric is the unique AC Ricci-flat Kähler metric with a smooth cone link. The work also constructs orbifold Kähler compactifications of $\mathbb{C}^n$, clarifies when these cones are Fano and Gorenstein, and, under Shokurov’s conjecture (known for $n\le 3$), yields an equivariant biholomorphism to $\mathbb{C}^n$ with a linear torus action. Overall, the paper advances a novel interplay between complex geometry, Sasaki geometry, and symplectic methods to classify compactifications and study metric rigidity.
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.