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McKay correspondence for symplectic quotient singularities. (English) Zbl 1060.14020

Let \(V\) be a finite dimensional complex vector space and \(G\subset \text{SL}(V)\) a finite subgroup. Let \(X:=V/G\). Then two natural questions arise: 1) when does \(X\) admit a crepant resolution of singularities \(f\colon Y\to X\), and 2) if such a resolution exists, what can be said about the homology \(H_*(Y,\mathbb Q)\)? In dimension \(2\) J. McKay proved that there always exists a crepant resolution \(f\colon Y\to X\) such that the fiber \(f^{-1}(0)\) over the singularity of \(X\) is a rational curve whose components are numbered by the conjugacy classes of \(G\); moreover, the homology classes of these components freely generate \(H_2(X,\mathbb Q)\) and \(H_i(X,\mathbb Q)=0\) for \(i>0\) and \(i\neq 2\). This is the so-called McKay correspondence [cf. J. McKay, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183–186 (1980; Zbl 0451.05026)]. If \(\text{dim}(V)=3\) the first question was solved affirmatively by several people independently, while the second question was solved by Y. Ito and M. Reid who proved that the same result holds true as in dimension \(2\) [in: Higher dimensional complex varieties. Proceedings of the International Conference, Trento, Italy, June 15–24, 1994, 221–240 (1996; Zbl 0894.14024)]. In the paper under review the author imposes an additional assumption of the pair \((V,G)\), namely he assumes that \(V\) has a nondegenerate symplectic form and the inclusions \(G\subset \text{Sp}(V)\subset \text{SL}(V)\) preserve not only the volume form in \(V\) but also the symplectic form. Under these hypotheses he proves a higher-dimensional analogue of the McKay correspondence.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14L30 Group actions on varieties or schemes (quotients)
14B05 Singularities in algebraic geometry
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