Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without -codimension-1 singularities. Let D be a suborbifold divisor in X such that and -pK (X) = q[D] for some 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 (Conlon and Hein, Duke Math J, 162:2855-2902, 2013), we show that each Kahler class on 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 (Invent Math, 106:27-60, 1991). (2) We introduce new methods to prove an analogous statement (with rate O(r (-0.0128))) when and is the strict transform of a smooth quadric through p in . Here D is no longer Kahler-Einstein, but the normal -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.