2013/01/22 by Ronan J. Conlon, Conlon, Ronan J., Hans‐Joachim Hein +1 · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1301.5312
openalex publication_date 2013/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D ⊃ Sing(X) and -pKX = q[D] for some p, q ∈ \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r-1-ε) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce new methods to prove an analogous statement (with rate O(r-0.0128)) when X = BlpP3 and D = Blp1,p2P2 is the strict transform of a smooth quadric through p in P3. Here D is no longer Kahler-Einstein, but the normal S1-bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.