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**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.

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.

Reviewer: Vladimir Guletskii (Minsk)