## Motivic integration, quotient singularities and the McKay correspondence.(English)Zbl 1080.14001

Let $$d$$ be a positive integer, $$k$$ be a field of characteristic zero containing all $$d$$-th roots of unity, and let $$G$$ be a finite subgroup of order $$d$$ in $$\text{SL}(k,n)$$ acting on the affine space $$\mathbb A^n_k$$. Consider a resolution $$Y\to X$$ of singularities on the quotient $$X=\mathbb A_k^n/G$$ and assume that $$Y$$ is crepant, i.e. $$K_Y=0$$. The McKay correspondence is a connection between irreducible representations of the group $$G$$ and cohomology of $$Y$$. In one form it says that the Euler number of $$Y$$ is equal to the number of conjugacy classes in $$G$$ [M. Reid, in: Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276, 53–72 (2002; Zbl 0996.14006)], and it was proved by V. Batyrev [J. Eur. Math. Soc. 1, No. 1, 5–33 (1999; Zbl 0943.14004)]. The present paper is devoted to a proof of the corresponding statement on the motivic level by means of motivic integration.
To be slightly more precise, let $$\mathcal M$$ be the Grothendieck group of algebraic varieties over $$k$$ with relation $$[X]=[Z]+[X-Z]$$ for Zariski closed $$Z$$ in $$X$$ and the product induced by products of varieties. Following the notation of the paper, let $$\mathcal M_{\text{loc}}$$ be the localization $$\mathcal M[\mathbb L^{-1}]$$, where $$\mathbb L=[\mathbb A^1]$$. Let also $$F^m\mathcal M_{\text{loc}}$$ be a subgroup generated by $$[X]\mathbb L^{-i}$$ with $$\dim (X)\leq i-m$$. The filtration $$F^m$$ gives the completion $$\hat {\mathcal M}$$ of $$\mathcal M_{\text{loc}}$$. At last, we add the relation $$[V/G]=[V]$$ for each $$k$$-vector space with a linear action of a finite group $$G$$ getting the corresponding quotient ring $$\hat \mathcal M_{/ }$$.
The above rings are closely connected with the category of Chow motives $$\text{CHM}_k$$ over $$k$$ with coefficients in $$\mathbb Q$$. Namely, there exists a function $$\chi _c$$ from the set of varieties over $$k$$ to $$K_0(\text{CHM}_k)$$ satisfying the nice properties listed on page 283 of the paper. The analogous filtration on $$K_0(\text{CHM}_k)$$ gives rise to the completion $$\hat K_0(\text{CHM}_k)$$. The map $$\chi _c$$ induces ring homomorphisms $$\chi _c:\mathcal M\to K_0(\text{CHM}_k)$$ and $$\hat \chi _c:\hat \mathcal M\to \hat K_0(\text{CHM}_k)$$, which can be factored through $$\mathcal M_{/ }$$ and $$\hat \mathcal M_{/ }$$ respectively.
Given a variety $$X$$ over $$k$$ let $$\mathcal L(X)$$ be the scheme of germs of arcs on $$X$$. For any field extension $$K/k$$ one has a natural bijection $$\mathcal L(X)(K)\cong \text{Hom}_k(K[[t]],X)$$ where $$K[[t]]$$ is the ring of formal power series with coefficients in $$K$$. If $$B^t$$ is a set of $$k[t]$$-semi-algebraic subsets in $$\mathcal L(X)$$, then there is a nice measure $$\mu :B^t\to \hat \mathcal M$$, called a motivic measure on $$\mathcal L(X)$$. Assume that $$X$$ is an irreducible normal variety of dimension $$n$$, which is Gorenstein with at most canonical singularities at each point. Some appropriate notion of integration with respect to $$\mu$$ with values in the ring $$\hat \mathcal M$$ gives rise to the notion of motivic Gorenstein measure $\mu^{\text{Gor}}(A)=\int _A{\mathbb L}^{-\text{ord}}_{t\omega _X}{\text{ d}}\mu$ of each subset $$A\in B^t$$. Here $$\text{ord}_t\omega _X$$ is the order of a global section $$\omega _X$$ of $$\Omega _X^n\otimes k(X)$$ generating $$\Omega _X^n$$ at each smooth point of $$X$$ (use that $$X$$ is Gorenstein with good singularities).
Now we are returning to the quotient $$X=\mathbb A^n_k/G$$, where $$G$$ is a finite subgroup of $$\text{SL}(k,n)$$. Let $$\mathcal L(X)_0$$ be the set of arcs whose origins are in the image of the point $$0$$ in the quotient $$X$$. The main result of the paper expresses the Gorenstein motivic measure $$\mu^{\text{Gor}}(\mathcal L(X)_0)$$ in terms of weights $$w(\gamma )$$ of conjugacy classes $$\gamma$$ in $$G$$. Namely, the equality $\mu ^{\text{Gor }}(\mathcal L(X)_0)= \sum _{\gamma \in \text{Conj}(G)}\mathbb L^{-w(\gamma )}$ holds in $$\hat \mathcal M_{/ }$$, where $$\text{Conj}(G)$$ is the set of conjugacy classes of $$G$$. As a corollary, if $$h:Y\to X$$ is a crepant resolution, then $[h^{-1}(0)]=\sum _{\gamma \in \text{Conj}(G)}\mathbb L^{n-w(\gamma )}$ in the ring $$\hat \mathcal M_{/ }$$. This is already a motivic expression of the McKay correspondence. If $$k=\mathbb C$$ and we pass to the Hodge realization, we get the result proved by Batyrev and conjectured by Reid.

### MSC:

 14E18 Arcs and motivic integration 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 14A15 Schemes and morphisms 14A20 Generalizations (algebraic spaces, stacks) 14L30 Group actions on varieties or schemes (quotients)

### Citations:

Zbl 0996.14006; Zbl 0943.14004
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