Classification of asymptotically conical Calabi–Yau manifolds

A Riemannian cone (C,gC) is by definition a warped product C= R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a Calabi–Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) 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.