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The McKay correspondence as an equivalence of derived categories. (English) Zbl 0966.14028
Summary: Let \(G\) be a finite group of automorphisms of a non-singular three-dimensional complex variety \(M\), whose canonical bundle \(\omega_M\) is locally trivial as a \(G\)-sheaf. We prove that the Hilbert scheme \(Y=G \text{-Hilb }{M}\) parametrising \(G\)-clusters in \(M\) is a crepant resolution of \(X=M/G\) and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on \(Y\) and coherent \(G\)-sheaves on \(M\). This identifies the K-theory of \(Y\) with the equivariant K-theory of \(M\), and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

MSC:
14J50 Automorphisms of surfaces and higher-dimensional varieties
18E30 Derived categories, triangulated categories (MSC2010)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
19L47 Equivariant \(K\)-theory
14L30 Group actions on varieties or schemes (quotients)
14J30 \(3\)-folds
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