Classification of asymptotically conical Calabi-Yau manifolds

2022/01/03 by Ronan J. Conlon, Conlon, Ronan J., Hans‐Joachim Hein +1 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2201.00870

openalex publication_date 2022/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Riemannian cone (C, g_C) is by definition a warped product C = ℝ⁺ × L with metric g_C = dr² ⊕ r² 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.

Cited by

• Complete Kähler manifolds with nonnegative Ricci curvature II
• Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones

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