Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Abstract

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \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-\epsilon}) 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 = Bl_{p}P^3 and D = Bl_{p_1,p_2}P^2 is the strict transform of a smooth quadric through p in P^3. Here D is no longer Kahler-Einstein, but the normal S^1-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.