Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Abstract

Let X be a compact Kähler orbifold without ${\mathbb{C}}$-codimension-1 singularities. Let D be a suborbifold divisor in X such that ${D \supset {\rm Sing}(X)}$ and −pK _X = q[D] for some ${p, q \in \mathbb{N}}$ with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kähler–Einstein, then, applying results from our previous paper (Conlon and Hein, Duke Math J, 162:2855–2902, 2013), we show that each Kähler class on ${X \setminus D}$ contains a unique asymptotically conical Ricci-flat Kähler 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 (Invent Math, 106:27–60, 1991). (2) We introduce new methods to prove an analogous statement (with rate O(r ^−0.0128)) when ${X = {\rm Bl}_p \mathbb{P}^{3}}$ and ${D = {\rm Bl}_{p_1,p_2} \mathbb{P}^{2}}$ is the strict transform of a smooth quadric through p in ${\mathbb{P}^3}$. Here D is no longer Kähler–Einstein, but the normal ${\mathbb{S}^1}$-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.