The \(p\)-cyclic McKay correspondence via motivic integration (Q2874606)

Let \(G\) be a finite subgroup of \(\mathrm{SL}_d(\mathbb{C})\) acting naturally on \(\mathbb{C}^d\). The classical McKay correspondence relates the representation theory of \(G\) to the geometry of a crepant resolution of \(\mathbb{C}^d/G\).NEWLINENEWLINEThe paper under review studies such a correspondence for a finite cyclic subgroup \(G\subset \mathrm{SL}_d(k)\) when the characteristic of the perfect field \(k\) is equal to \(|G|=p\). This is the special case of the so called wild McKay correspondence. For a finite dimensional \(G\)-representation \(V\), let \(V=\bigoplus_{\lambda=1}^l V_{d_\lambda}\) be the decomposition into indecomposable representations where the index \(d_\lambda\) denotes the dimension of \(V_{d_\lambda}\). Let \(D_V=\sum_{\lambda=1}^l d_\lambda(d_\lambda-1)/2\). For \(D_V\geq p\) the stringy motivic invariant of the quotient variety \(X=V/G\) denoted by \(M_{\mathrm{st}}(X)\) is defined by means of a motivic integration over the arc space of \(X\). The main result of the paper under review proves the following the formula NEWLINE\[NEWLINEM_{\mathrm{st}}(X)=\mathbb{L}^d+\frac{\mathbb{L}^{l-1}(\mathbb{L}-1)(\sum_{s=1}^{p-1}\mathbb{L}^{s-\sum_{\lambda=1}^l\sum_{i=1}^{d_\lambda-1}[ij/p]})}{1-\mathbb{L}^{p-1-D_V}}.NEWLINE\]NEWLINE If \(X\) admits a crepant resolution of singularities \(Y\) then \(D_V=p\) and this gives a simple formula for \([Y]=M_{\mathrm{st}}(X)\) by the virtue of a result of \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 1--32 (1998; Zbl 0963.14015)]. In particular this shows that \(Y\) has topological Euler characteristic \(p\) as conjectured by Reid and proven by \textit{V. V. Batyrev} [J. Eur. Math. Soc. (JEMS) 1, No. 1, 5--33 (1999; Zbl 0943.14004)] in characteristic 0. The paper under review also proves a Poincaré duality for the stringy motivic invariant of the projectivization of \(X\). The proofs use the the motivic integration over the space of twisted arcs in quotient stacks.