CLASSIFICATION OF ASYMPTOTICALLY CONICAL CALABI-YAU MANIFOLDS

A Riemannian cone (C, gC) is by definition a warped product C = RC x L with boundary. We say that C is a Calabi-Yau cone if gC is a Ricci-flat K & auml;hler 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-K & auml;hler 4-manifolds without twistor theory.