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

TL;DR

This paper characterizes certain Kähler compactifications of complex Euclidean spaces, proving a biholomorphic equivalence to standard projective space pairs and analyzing Ricci-flat metrics with specific asymptotic properties.

Contribution

It establishes a biholomorphic classification of Kähler compactifications of ^n and introduces a new method to characterize minimal discrepancy of Fano cone singularities.

Findings

^n with a Ka4hler submanifold complement is biholomorphic to (P^n, P^{n-1})

The flat metric on ^3 is unique among asymptotically conical Ricci-flat Ka4hler metrics with smooth cone at infinity

New characterization of minimal discrepancy using S^1-equivariant positive symplectic homology.

Abstract

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, as an application, 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.