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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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