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McKay correspondence and Hilbert schemes in dimension three. (English) Zbl 0995.14001
From the abstract: Let \(G\) be a nontrivial finite subgroup of \(SL_n (\mathbb{C})\). Suppose that the quotient singularity \(\mathbb{C}^n/G\) has a crepant resolution \(\pi:X\to \mathbb{C}^n/G\) (i.e. \(K_X= {\mathcal C}_X)\). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of \(X\) and the representations (or conjugacy classes) of \(G\) with a “certain compatibility” between the intersection product and the tensor product. The purpose of this paper is to give more precise formulation of the conjecture when \(X\) can be given as a certain variety associated with the Hilbert scheme of points in \(\mathbb{C}^n\). We give the proof of this new conjecture for an abelian subgroup \(G\) of \(SL_3(\mathbb{C})\).

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
14C05 Parametrization (Chow and Hilbert schemes)
14M17 Homogeneous spaces and generalizations
19E08 \(K\)-theory of schemes
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