Motivic multiplicative McKay correspondence for surfaces

TL;DR

This paper proves a motivic version of the McKay correspondence for surfaces, establishing an isomorphism between the Chow motives of minimal resolutions and orbifold motives, confirming the motivic crepant resolution conjecture in two dimensions.

Contribution

It introduces a motivic framework incorporating the orbifold product for the McKay correspondence on surfaces, extending previous cohomological results to Chow motives.

Findings

Isomorphism of algebra objects in Chow motives category

Equivalence of Chow rings, Grothendieck rings, and cohomology rings

Confirmation of the two-dimensional Motivic Crepant Resolution Conjecture

Abstract

We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture.