A Construction of Complete Ricci-flat Kähler Manifolds

Abstract

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat Kähler metrics on quasi-projective varieties Y=X\D with α[D]=c_1(X), α>1. The requirement that D admit a Kähler-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 Kähler metric asymptotic to a Calabi ansatz metric. Some examples are given which were not previously known.

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