We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H 2 c (Y, R). A Kähler cone (X,ḡ) is a metric cone over a Sasaki manifold (S, g), i.e. X = C(S) := S × R >0 withḡ = dr 2 + r 2 g, and (X,ḡ) is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat Kähler metrics on crepant resolutions π : Y → X = C n / , with ⊂ SL(n, C), due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric Kähler cone admits a Ricci-flat Kähler cone metric. It follows that if a toric Kähler cone X = C(S) admits a crepant resolution π : Y → X , then Y admits a T n -invariant Ricci-flat Kähler metric asymptotic to the cone metric (X,ḡ) in every Kähler class in H 2 c (Y, R). A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X . We then list some examples which are easy to construct using toric geometry.

Previously the author has proved that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H 2 c (Y, R). These manifolds can be considered to be generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric Kähler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat Kähler cone metrics by the work of C. Boyer, K. Galicki, J. Kollár, and others. We concentrate on 3-dimensional examples. Two families of hypersurface examples are given which are distinguished by the condition b 3 (Y ) = 0 or b 3 (Y ) = 0.

Abstract. We consider the stability of Sasaki-extremal metrics under de-

Extending the results of S. Y. Cheng and S.-T. Yau it is shown that a strictly pseudoconvex domain M ⊂ X in a complex manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ∂M is normal. In this case M must be a domain in a resolution of the Sasaki cone over ∂M . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kähler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities.We give many examples of Kähler-Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c 1 < 0 admits a complete Kähler-Einstein metric.

We define K-stability of a polarized Sasakian manifold relative to a maximal torus of automorphisms. The existence of a Sasaki-extremal metric in the polarization is shown to imply that the polarization is K-semistable. Computing this invariant for the deformation to the normal cone gives an extention of the Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an obstruction of Sasaki-extremal metrics. We use this to give a list of examples of Sasakian manifolds whose Sasaki cone contains no extremal representatives. These give the first examples of Sasaki cones of dimension greater than one that contain no extremal Sasaki metrics whatsoever. In the process we compute the unreduced Sasaki cone for an arbitrary smooth link of a weighted homogeneous polynomial.1991 Mathematics Subject Classification. 53C25 primary, 32W20 secondary.