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 .

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Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) × X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M ; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M ) = SU(n), where n is the complex dimension of M . If n > 2 we then show that there exists a projective orbifold M and a divisor D ∈ |−K M | with torsion normal bundle such that M is biholomorphic to M \ D, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M , D) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M \ D.

Abstract. One of the main results of the paper [6] by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kähler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.

Let X be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let L be an ample line bundle on X. Assume that the pair (X, L) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point x ∈ X there exist a Kähler-Einstein Fano manifold Z and a positive integer q dividing KZ such that − 1 q KZ is very ample and such that the germ (X, x) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of 1 q KZ. We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing 2πc1(L) on X is asymptotic at a polynomial rate near x to the natural Ricci-flat Kähler cone metric on 1 q KZ constructed using the Calabi ansatz. In particular, our result applies if (X, O(1)) is a nodal quintic threefold in P 4 . This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.