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

Abstract

Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $\pi: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is the cone $C(S)=\Bbb R_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall prove that the crepant resolution $X$ of $X_0$ admits a complete Ricci-flat K\"ahler metric in every K\"ahler class in $H^2(X)$. We apply the continuity method for solving the Monge-Amp\`ere equation to obtain a relevant existence theorem and a uniqueness theorem of Ricci-flat conical K\"ahler metrics. By using the vanishing theorem on the crepant resolution $X$ and the Hodge and Lefschetz decompositions of the basic cohomology groups on the Sasakian manifold $S$, we construct an initial K\"ahler metric in every K\"ahler class on which the existence theorem can be applied.We show there are many examples of Ricci-flat complete K\"ahler manifolds arising as crepant resolutions.