Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Original · EN

Let X be a compact Kahler orbifold without -codimension-1 singularities. Let D be a suborbifold divisor in X such that D ⊃ Sing(X) and -pKₓ = q[D] for some p, q ∈ 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 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.