Mathematics

In many cases, a crepant resolution of $\mathbb{C}^3/G$, where $G$ is a finite subgroup of  $SL(3,\mathbb{C})$, happens to be the total space of the canonical bundle over a Kahler manifold of complex dimension $2$. Since any such crepant resolution is a noncompact Calabi-Yau, there is a natural question of constructing a Ricci-flat metric on it. The existence of such metrics was proven by Joyce. In this talk, I will show how to construct a Ricci-flat metric when a crepant resolution is a space $X=tot(K_{M})$, where $K_{M}$ is the canonical bundle over a 2-dimensional Kahler manifold $M$.