The higher-dimensional McKay correspondence for crepant desingularizations

Let $G$ be a finite subgroup of $\operatorname{SL}(r,\mathbb{C})$, let $X=\mathbb{C}^{r}/G$, and let $\widehat{X}\longrightarrow X$ be a crepant desingularization. For each integer $i$, let $Z_g$ be the Zariski closure in $\widehat X$ of the center of the monomial valuation of the function field of $X$ corresponding to a conjugacy class $[g]$ of age $i$. Higher-dimensional McKay correspondence. There is a canonical one-to-one correspondence

$$\left\{\text{conjugacy classes of }G\text{ having age }i\right\}\longleftrightarrow H_{2(r-i)}^{\operatorname{BM}}(\widehat X,\mathbb{Q}),$$

which maps $[g]$ to the fundamental class of the algebraic cycle $Z_g$. This is the expected direct analogue of the McKay correspondence over $\mathbb{Q}$ in dimensions at least four; the existence of the required crepant desingularization and an integral version remain unclear in general.