Classification of asymptotically conical Calabi–Yau manifolds

Abstract

A Riemannian cone $(C,g_C)$ is by definition a warped product $C={\mathbb{R}}^{+}\times L$ with metric $g_C=d{r}^2\oplus r^2{g}_L$, 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 Kähler 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-Kähler 4-manifolds without twistor theory.