Classification of asymptotically conical Calabi-Yau manifolds

TL;DR

This paper classifies all smooth complete Calabi-Yau manifolds that asymptotically resemble a given Calabi-Yau cone, providing a comprehensive understanding of their structure and including a proof of Kronheimer's classification of ALE hyper-K"ahler 4-manifolds.

Contribution

It offers a complete classification of asymptotically conical Calabi-Yau manifolds and includes a proof of Kronheimer's classification of ALE hyper-K"ahler 4-manifolds.

Findings

Classified all smooth complete Calabi-Yau manifolds asymptotic to a Calabi-Yau cone.

Proved Kronheimer's classification of ALE hyper-K"ahler 4-manifolds.

Established polynomial rate asymptotics at infinity.

Abstract

A Riemannian cone $(C, g_C)$ is by definition a warped product $C = \mathbb{R}^+ \times L$ with metric $g_C = dr^2 \oplus r^2 g_L$, where $(L,g_L)$ is a compact Riemannian manifold without boundary. We say that $C$ is a Calabi-Yau cone if $g_C$ is a Ricci-flat K\"ahler metric and if $C$ admits a $g_C$-parallel holomorphic volume form; this is equivalent to the cross-section $(L,g_L)$ being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-K\"ahler $4$-manifolds without twistor theory.