Asymptotically conical Calabi-Yau manifolds, III

TL;DR

This paper proves that certain asymptotically conical Calabi-Yau manifolds can be compactified into projective algebraic varieties, enabling classification and construction results including Kronheimer's ALE spaces and Stenzel's metric.

Contribution

It demonstrates that Calabi-Yau manifolds with asymptotically conical geometry admit projective algebraic compactifications and provides classification theorems via deformation techniques.

Findings

Calabi-Yau manifolds admit projective algebraic compactifications.

Classification theorems for asymptotically conical Calabi-Yau manifolds.

Includes classification of ALE spaces and uniqueness of Stenzel's metric.

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.