A Construction of Complete Ricci-flat K\"ahler Manifolds

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat K"{a}hler metrics on quasi-projective varieties Y=X\D with \alpha[D]=c_1(X), \alpha >1. The requirement that D admit a K"{a}hler-Einstein metric is generalized to the condition that the link S in the normal bundle of D admits a Sasaki-Einstein structure in the Sasaki-cone of the usual Sasaki structure provided the embedding D\subset X satisfies an additional holomorphic condition. If D is a toric variety, then S always admits a Sasaki-Einstein metric. As an application we prove that every small smooth deformation of a toric Gorenstein singularity admits a complete Ricci-flat K"{a}hler metric asymptotic to a Calabi ansatz metric. Some examples are given which were not previously known.