Kavli Affiliate: Yukari Ito
| Summary:
Let $G$ be a nontrivial finite subgroup of $SL_n(C)$. Suppose that the quotient singularity $C^n/G$ has a crepant resolution $πcolon Xto C^n/G$ (i.e. $K_X = shfO_X$). 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 (see e.g. citeMaizuru). 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 $C^n$. We give the proof of this new conjecture for an abelian subgroup $G$ of $SL_3(C)$.
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