Ronan J. Conlon, Hans-Joachim Hein
Duke Math. J. Advance Publication, 1-69, (2024) DOI: 10.1215/00127094-2023-0030
KEYWORDS: Calabi-Yau manifolds, asymptotically conical Calabi-Yau, Euclidean volume growth, Tian-Yau construction, Kronheimer’s classification, deformations and resolutions of isolated singularities, deformation to the normal cone, 53C25, 14J32
A Riemannian cone is by definition a warped product with metric , where is a compact Riemannian manifold without boundary. We say that C is a Calabi–Yau cone if is a Ricci-flat Kähler metric and if C admits a -parallel holomorphic volume form; this is equivalent to the cross-section 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.