Asymptotically conical Calabi-Yau manifolds, III

Abstract

In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show that if the open manifold is Calabi-Yau, then Chi Li's compactification is projective algebraic. This has two applications. First, every Calabi-Yau manifold of this kind can be constructed using our refined Tian-Yau type theorem from the second article in this series. Secondly, we prove classification theorems for such manifolds via deformation to the normal cone. This includes Kronheimer's classification of ALE spaces and a uniqueness theorem for Stenzel's metric.

added more details concerning deformation to the normal cone and fixed a gap in the argument (normality of the central fibre)