Ricci-flat Kahler metrics on crepant resolutions of Kahler cones

We prove that a crepant resolution pi : Y -> X of a Ricci-flat Kahler cone X admits a complete Ricci-flat Kahler metric asymptotic to the cone metric in every Kahler class in H-c(2) (Y, R). A Kahler cone (X, (g) over bar) is a metric cone over a Sasaki manifold (S, g), i.e. X = C(S) := S x R->0 with (g) over bar = dr(2) + r(2)g, and (X, (g) over bar ) is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat Kahler metrics on crepant resolutions pi : Y -> X = C-n/Gamma, with Gamma subset of 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 K hler cone admits a Ricci-flat K hler cone metric. It follows that if a toric Kahler cone X = C(S) admits a crepant resolution pi : Y -> X, then Y admits a T-n-invariant Ricci-flat K hler metric asymptotic to the cone metric (X, (g) over bar ) in every K hler class in H-c(2) (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.