Let X be a smooth complex manifold. Assume that Y subset of X is a K & auml;hler submanifold such that X\Y is biholomorphic to Cn. We prove that (X,Y) is biholomorphic to (Pn,Pn-1). We then study certain K & auml;hler orbifold compactifications of Cn and, as an application, prove that on C3 the flat metric is the only asymptotically conical Ricci-flat K & auml;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 S1-equivariant positive symplectic homology.

Publication:

INVENTIONES MATHEMATICAE

http://dx.doi.org/10.1007/s00222-025-01377-2

Author:

Zhengyi Zhou

State Key Laboratory of Mathematical Sciences, Chinese Academy of Sciences, Beijing, China

Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, China

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

zhyzhou@amss.ac.cn