Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities

Abstract

Let X₀ be an affine variety with only normal isolated singularity at p. We assume that the complement X₀ \ {p} is biholomorphic to the cone C(S) of an Einstein-Sasakian manifold S of real dimension 2n − 1. If there is a resolution of singularity π: X → X₀ with trivial canonical line bundle K_X, then there is a Ricci-flat complete Kähler metric for every Kähler class of X. We also obtain a uniqueness theorem of Ricci-flat conical Kähler metrics in each Kähler class with a certain boundary condition. We show there are many examples of Ricci-flat complete Kähler manifolds arising as crepant resolutions.