Ricci-flat K\"ahler metrics on crepant resolutions of K\"ahler cones

TL;DR

This paper proves the existence of complete Ricci-flat K"ahler metrics on crepant resolutions of Ricci-flat K"ahler cones, including toric cases, using geometric analysis and toric geometry techniques.

Contribution

It establishes the existence of Ricci-flat K"ahler metrics on crepant resolutions of K"ahler cones, extending known results to toric and more general cases.

Findings

Existence of Ricci-flat K"ahler metrics on crepant resolutions.

Construction of such metrics using toric geometry.

Application to ALE Ricci-flat K"ahler metrics on quotient singularities.

Abstract

We prove that a crepant resolution of a Ricci-flat K\"ahler cone X admits a complete Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\"ahler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat K\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.