Abstract. This is the first part in a series of articles on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition O(r −n−ε ) needed in earlier work to O(r −ε ), relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on T * S n is −2 n n−1 .

We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation.

For each n ≥ 3, we construct on C n examples of complete Calabi-Yau metrics of Euclidean volume growth having a tangent cone at infinity with singular cross-section.

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Let X be a compact Kähler 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 Kähler-Einstein, then, applying results from our previous paper [15], we show that each Kähler class on X \ 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 [54]. (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128 )) when X = BlpP 3 and D = Blp 1 ,p 2 P 2 is the strict transform of a smooth quadric through p in P 3 . Here D is no longer Kähler-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.Date: July 4, 2018. 1 arXiv:1301.5312v3 [math.DG] 8 Dec 2014 Comparing the two theorems and their proofs. Many of the refinements in Theorem A (no neat or almost ample condition, all Kähler classes, the parameter c, uniqueness and symmetry) are due to an improvement of general technique in [15], partly based on important earlier contributions by van Coevering [55], whereas asymptotics of the form (1.1) are already implicit in Tian-Yau [54]. Let us point out one useful consequence of our explicit estimate (1.2).Corollary B. If X is smooth, then the best possible convergence rate λ of the Ricci-flat metrics of Theorem A to their tangent cones at infinity is always strictly greater than 1.Proof. Since N q−p D = K −p D and Pic(D) is torsion-free because π 1 (D) = 0, there exists a line bundle L with L p = N D and L q−p = K −1 D . Thus, by [32, p. 32, Corollary], q − p n with equality if and only if D = P n−1 , so that α − 1 n and λ 1 with equality if and only if D = P n−1 , N D = O(1). But in the latter case, the cone, and hence (X \ D, g c ) itself, must be isometric to flat C n .It seems reasonable to expect that α − 1 n even if X is singular. Moreover, equality should still imply that the cone is C n /Γ (see Remark A.2 for some examples where Γ = {1}), so that λ 2n by 1 Cristiano Spotti pointed out to us that this phenomenon was first observed in [41] for the classical elliptic modular curve H/PSL(2, Z), which is isomorphic to C as a variety but whose π orb 1 and Pic orb are nontrivial.